In △ABC, D is the midpoint of BC and E is the midpoint of AD. If BE and CE are connected, and the area of △ABC is 20, then the area of △BCE is (). option:['5', '10', '15', '18']

As per the question, the image is represented below:

Since D is the midpoint of BC, we have the area of triangle ABD = the area of triangle ACD = 0.5 × the area of triangle ABC = 10. As E is the midpoint of AD, the area of triangle EBD = 0.5 × the area of triangle ABD = 5, and the area of triangle ECD = 0.5 × the area of triangle ACD = 5. Therefore, the area of triangle BCE = the sum of the areas of triangle EBD and ECD = 10.

The final answer is: 10


In △ABC, points D and E are the midpoints of AC and AB, respectively. And the area of △ABC is 12. Connecting BD and DE, the area of △BDE is (). option:['12', '6', '3', '2']

Adhering to the provided question, the picture is presented as follows:

Since point D is the midpoint of AC, we have S△ABD = 0.5×S△ABC = 0.5×12 = 6. Since point E is the midpoint of AB, we have S△BDE = 0.5×S△ABD = 0.5×6 = 3.

The final answer is: 3


AD is the median of △ABC, E is the midpoint of AD, the area of △ABE is 2, the area of △ABC is (). option:['5', '6', '7', '8']

Taking into consideration the question, the image is illustrated below:

Since AD is the median of triangle ABC, the areas of triangles ABD and ACD are equal to half of the area of triangle ABC.
Furthermore, since BE is the median of triangle ABD, the areas of triangles ABE and DBE are equal to half of the area of triangle ABD, which is equal to 2.
Therefore, the area of triangle ABD is 4, and the area of triangle ABC is 8.

The final answer is: 8


In Rt△ABC, ∠ACB is equal to 90°. AC is 3 and BC is 4. Use point A as the center, draw an arc with AC as the radius, intersecting AB at point D. The length of BD is (). option:['2', '3', '4', '5']

Based on the given question, the picture is illustrated below:

Since AC = 3 and BC = 4, we have AB = √{AC²+BC²} = √{3²+4²} = 5. Since we draw an arc with point A as the center and AC as the length radius, intersecting with AB at point D, we have AD = AC. Therefore, AD = 3 and BD = AB - AD = 5 - 3 = 2.

The final answer is: 2


In △ABC, AB is perpendicular to BC and AC = 2.5, AB = 1. P is an arbitrary point on BC. The possible length of line segment AP is (). option:['0.5', '0.7', '2.3', '2.8']

∵P is any point on BC, therefore AB ≤ AP ≤ AC, which means 1 ≤ AP ≤ 2.5.

The final answer is: 2.3


In right triangle △ABC, ∠C is equal to 90°. AC is equal to 5 and BC is equal to 12. The measure of AB is (). option:['12', '13', '14', '15']

In Rt△ABC, ∠C = 90°. According to the Pythagorean theorem, we have: AB = √{AC²+BC²} = √{5²+12²} = 13.

The final answer is: 13


In △ABC, ∠ACB is equal to 90°, AC equals 8, AB equals 10. DE perpendicular bisects AC and intersects AB at point E and intersects AC at point D. The length of DE is (). option:['3', '4', '5', '6']

Abiding by the question, the image is depicted below:

In △ABC, ∠ACB = 90°. By the Pythagorean theorem, we have: BC = √{AB²-AC²} = √{10²-8²} = 6. Since DE bisects AC, we get AD = 0.5×=4, ∠ADE = 90°. Therefore, tanA = \frac{BC}{AC} = \frac{DE}{AD}, so \frac{6}{8} = \frac{DE}{4}, hence DE = 3.

The final answer is: 3


In △ABC, ∠ACB is equal to 90°, AC has a length of 6, and BC has a length of 8. The length of the altitude CD on side AB is (). option:['4', '\\frac{24}{5}', '3√{3}', '10']

In triangle ABC, ∠ACB = 90°, AC = 6, BC = 8, then by the Pythagorean theorem: AB = √{AC²+BC²} = √{6²+8²} = 10. Since S△ABC = 0.5×AB•CD = 0.5×AC•BC, therefore CD = \frac{AC×BC}{AB} = \frac{6×8}{10} = \frac{24}{5}.

The final answer is: \frac{24}{5}


In △ABC, ∠C = 90°, point D is on side BC, AD = BD, DE bisects ∠ADB and intersects AB at point E. If AC = 12, BC = 16, then the length of AE is () option:['6', '8', '10', '12']

Following the given question, here is the image we've created:

In △ABC, ∠C = 90°, AC = 12, BC = 16. By the Pythagorean theorem, we know: AB = √{AC² + BC²} = √{12² + 16²} = 20. Since AD = BD, DE bisects ∠ADB intersecting AB at point E. Therefore, AE = BE = 0.5×AB = 10.

The final answer is: 10


Given △ABC, AB = 10, AC = 8, BC = 6, the perpendicular bisector of AC intersects AB at point D, with the foot E, and CD is connected. The length of CD is (). option:['3', '4', '4.8', '5']

Since AB = 10, AC = 8, BC = 6, we have BC² + AC² = AB². Therefore, triangle ABC is a right triangle. Since DE is the perpendicular bisector of AC, we have AE = EC = 4. DE is parallel to BC and segment DE is the median of triangle ABC. Therefore, DE = 3. Thus, AD = DC = √(AE² + DE²) = √(3² + 4²) = 5.

The final answer is: 5


In △ABC, CD is the altitude on side AB. If AB = 1.5, BC = 0.9, and AC = 1.2, then the value of CD is() option:['0.72', '2.0', '1.125', 'uncertain']

Since AB = 1.5, BC = 0.9, AC = 1.2, we have AB² = 1.5² = 2.25, BC² + AC² = 0.9² + 1.2² = 2.25. Therefore, AB² = BC² + AC². Therefore, ∠ACB = 90°. Since CD is the altitude from AB, we have the area of triangle ABC is 0.5 × AB × CD = 0.5 × AC × BC. Therefore, 1.5CD = 1.2 × 0.9, CD = 0.72.

The final answer is: 0.72


OC bisects ∠AOB, point P is a point on OC, and PM is perpendicular to OB at point M. Point N is a moving point on ray OA. If OM=4 and OP=5, the minimum value of PN is () option:['2', '3', '4', '5']

Since PM is perpendicular to OB at point M, OM = 4, and OP = 5, we have PM = 3. When PN is perpendicular to OA, the value of PN is the smallest. Since OC bisects ∠AOB, PM is perpendicular to OB, so PM = PN. Since PM = 3, the minimum value of PN is 3.

The final answer is: 3


In △ABC, AD is perpendicular to BC at D, AB = 17, BD = 15, and DC = 6. The length of AC is () option:['11', '10', '9', '8']

Since AD ⊥ BC, we have ∠ADB = ∠ADC = 90°. Furthermore, given AB = 17, BD = 15, DC = 6, in the right triangle ABD, by the Pythagorean theorem we have: AD² = AB² - BD² = 64. In the right triangle ACD, by the Pythagorean theorem we have: AC = √{AD² + CD²} = √{8² + 6²} = 10, thus AC = 10.

The final answer is: 10


O is the midpoint of the diagonal AC of rectangle ABCD, and M is the midpoint of AD. If AB = 5 and AD = 12, the perimeter of quadrilateral ABOM is (). option:['16', '20', '29', '34']

As per the question, the image is represented below:

∵AB = 5, ∴CD = 5, ∵AD = 12, ∠D = 90°, ∴AC = 13. Since O and M are the midpoints of AC and AD respectively, ∴OB = 6.5, AM = 0.5 × AD = 6, OM is the median of triangle ACD, ∴OM = 0.5 × CD = 2.5, ∴The quadrilateral ABOM is S = AB + BO + OM + MA = 5 + 6.5 + 2.5 + 6 = 20.

The final answer is: 20


In right triangle △ABC, with ∠BAC equal to 90°, points D, E, and F are respectively the midpoints of the three sides, and DE is equal to 4cm. The length of AF is (). option:['2cm', '3cm', '4cm', '6cm']

Since point D and point E are the midpoints of AB and AC, respectively, we have DE as the median of triangle ABC. Therefore, BC = 2DE = 8cm. In right triangle BAC, point F is the midpoint of the hypotenuse BC, so AF = 0.5×BC = 4cm.

The final answer is: 4cm


Quadrilateral ABCD is a rhombus. Diagonals AC and BD intersect at point O. E is the midpoint of AD. If OE=3.5, then the perimeter of rhombus ABCD is () option:['14', '28', '7', '35']

Since quadrilateral ABCD is a rhombus, we have AB = BC = CD = AD. AC is perpendicular to BD. Since E is the midpoint of AD, OE is the median of right triangle AOD's hypotenuse. Therefore, AD = 2OE = 7. Hence, the perimeter of rhombus ABCD is 4x7 = 28.

The final answer is: 28


In rhombus ABCD, diagonal AC and BD intersect at point O. Point M is the midpoint of side AB. If MO = 4 cm, then the perimeter of rhombus ABCD is () option:['32cm', '24cm', '16cm', '8cm']

According to the given question, the image is depicted as follows:

Since quadrilateral ABCD is a rhombus, we have BO = DO, meaning O is the midpoint of BD. Also, since M is the midpoint of AB, MO is the median of triangle ABD. Therefore, AD = 2MO = 2 × 4 = 8 (cm), hence the perimeter of rhombus ABCD = 4AD = 4 × 8 = 32 (cm).

The final answer is: 32cm


In parallelogram ABCD, diagonals AC and BD intersect at point O. OE is parallel to AB and intersects AD at point E. If OA=2 and the perimeter of △AOE is 10, then the perimeter of parallelogram ABCD is (). option:['16', '32', '36', '40']

As dictated by the question, the image is shown as follows:

Since the quadrilateral ABCD is a parallelogram, we have AB = CD, AD = BC, OB = OD. Given OE || AB, OE is the median of triangle ABD. Therefore, AB = 2OE, AD = 2AE. Since the perimeter of triangle AOE is 10, we have OA + AE + OE = 10, and AE + OE = 5. So OA = 10 - 2 = 8, AB + AD = 2AE + 2OE = 16. Thus, the perimeter of square ABCD = 2 * (AB + AD) = 2 * 16 = 32.

The final answer is: 32


In right triangle △ABC, ∠ACB is equal to 90°. Points H, E, and F are the midpoints of sides AB, BC, and CA, respectively. Connecting EF and CH, if EF + CH = 4, the value of CH is () option:['1.5', '2', '2.5', '3']

Adhering to the provided question, the picture is presented as follows:

Since points E and F are the midpoints of sides BC and CA, we have EF = 0.5 × AB. In right triangle ABC with ∠ACB = 90°, point H is the midpoint of side AB, so CH = 0.5 × AB. Therefore, EF = CH, and since EF + CH = 4, CH = 2.

The final answer is: 2


The perimeter of ▱ABCD is 36. Diagonals AC and BD intersect at point O. Point E is the midpoint of CD. The perimeter of △DOE is 15. The length of BD is (). option:['18', '16', '14', '12']

Since quadrilateral ABCD is a parallelogram with a perimeter of 36, diagonals AC and BD intersect at point O. Therefore, AD + CD = 18, BD = 2OD. Point E is the midpoint of CD, so OE is the median of triangle ACD, DE = 0.5 × CD. Thus, OE = 0.5 × AD. Since the perimeter of triangle DOE is 15, OD + OE + DE = 15 leads to OD + 0.5 × AD + 0.5 × CD = 15, OD + 9 = 15, OD = 6, so BD = 12.

The final answer is: 12


In equilateral △ABC, points D and E are the midpoints of sides AB and BC, respectively. If DE = 2, then the perimeter of △ABC is () option:['9', '12', '16', '18']

In accordance with the provided context, the image is presented below:

Since D and E are the midpoints of sides AB and BC, DE = 2. Therefore, DE is the median of triangle ABC, so DE is parallel to AC and 2DE = AC = 4. Since triangle ABC is an equilateral triangle, the perimeter of triangle ABC = 3AC = 12.

The final answer is: 12


In parallelogram ABCD, the diagonals AC and BD intersect at point O. E is the midpoint of AD. Connect OE. If OE = 3, the length of CD is (). option:['3', '5', '6', '9']

Following the question, we present the image as follows:

∵Quadrilateral ABCD is a parallelogram, therefore AO = CO, and since E is the midpoint of AD, EO is the median of △ACD, thus CD = 2EO = 6.

The final answer is: 6


In rhombus ABCD, AC = 6, BD = 8, AH ⊥ BC at point H, then AH = () option:['24', '10', '\\frac{24}{5}', '\\frac{12}{5}']

Diagonal AC and BD intersect at point O. Since quadrilateral ABCD is a rhombus, we have AC ⊥ BD, OA = OC, and OB = OD. Therefore, BC = √{OC²+OB²} = √{3²+4²} = 5. Since the area of rhombus ABCD = 0.5 × AC × BD = 0.5 × 6 × 8 = 24, we have AH = 24/5.

The final answer is: \frac{24}{5}


In ▱ABCD, the angle bisector of ∠ABC intersects AD at point E, and the angle bisector of ∠BCD intersects AD at point F. If AB = 4 and AF = 1, then the length of BC is (). option:['4', '5', '7', '6']

Since the quadrilateral ABCD is a parallelogram, we have AD ∥ CB, AB = CD = 4, and AD = BC. Therefore, ∠DFC = ∠FCB. Furthermore, since CF bisects ∠BCD, we have ∠DCF = ∠FCB. Therefore, ∠DFC = ∠DCF, and DF = DC = 4. As AF = 1, we have AD = 4 + 1 = 5, so BC = 5.

The final answer is: 5


In △ABC, points D and E are the midpoints of BC and AC, respectively. BF bisects ∠ABC, intersecting DE at point F. If BC = 6, the length of DF is (). option:['2', '3', '6', '4']

As per the question, the image is represented below:

Since D and E are respectively the midpoints of BC and AC, we have DE∥AB. Therefore, ∠BFD = ∠ABF. Since BF bisects ∠ABC, we have ∠DBF = ∠ABF. Thus, ∠BFD = ∠DBF, and then DF = DB = 0.5×BC = 0.5×6 = 3.

The final answer is: 3


In diamond ABCD, AB=5, diagonals AC and BD intersect at point O, and AC=6. AE is perpendicular to CD at point E. The length of AE is () option:['4', '\\frc{12}{5}', '\\frac{24}{5}', '5']

Since quadrilateral ABCD is a diamond, we have AO = 0.5 × AC = 0.5 × 6 = 3, OB = 0.5 × BD, AC ⊥ BD. Since AB = 5, thus BO = √{AB²-AO²} = √{5²-3²} = 4. Hence, BD = 8. The area of diamond ABCD is 0.5 × AC•BD = CD•AE. Therefore, 0.5 × 6 × 8 = 5AE, and thus AE = 24/5.

The final answer is: \frac{24}{5}


In quadrilateral ABCD, AD is parallel to BC, AD = 2, BC = 5. Connect AC, BD, points E and F are the midpoints of diagonals AC and BD respectively. Connect EF, then the length of EF is () option:['1', '1.5', '2.5', '3.5']

Since the midpoint of DC is G, connect FG and EG. Since E and F are the midpoints of the diagonals AC and BD, we have FG ∥ BC and EG ∥ AD. Since AD ∥ BC, EG ∥ BC, FG ∥ EG, so E, F, and G are collinear. Therefore, FG is the median of △BCD, so FG = 0.5×BC = 2.5. Since AD ∥ BC, EG ∥ AD, EG is the median of △ACD, so EG = 0.5×AD = 1, and EF = FG - EG = 1.5.

The final answer is: 1.5


In ▱ABCD, BC is equal to 10, AC is equal to 8, and BD is equal to 14. Points AC and BD intersect at point O. The perimeter of △AOD is (). option:['16', '20', '21', '23']

According to the given question, the image is depicted as follows:

Since quadrilateral ABCD is a parallelogram, we have AB = CD, BC = AD = 10, AO = CO = 0.5×AC = 4, BO = DO = 0.5×BD = 7. Therefore, the perimeter of triangle AOD is: AD + AO + DO = 10 + 4 + 7 = 21.

The final answer is: 21


Given △ABC, points D, E, F are the midpoints of sides BC, CA, AB, respectively. AB = 10, AC = 8. Connect DE, EF, FD. The perimeter of quadrilateral AFDE is equal to () option:['18', '16', '14', '12']

Since D, E, and F are the midpoints of the sides BC, CA, and AB respectively. AB = 10, AC = 8, therefore DE = 0.5 × AB = 5, DF = 0.5 × AC = 4, AF = 0.5 × AB = 5, AE = 0.5 × AC = 4. Therefore, the perimeter of quadrilateral AFDE is equal to AF + DF + DE + AE = 5 + 5 + 4 + 4 = 18.

The final answer is: 18


In the parallelogram ABCD, diagonals AC and BD intersect at point O. If AC = 12, BD = 16, and CD = 11, the perimeter of △DOC is (). option:['39', '31', '33', '25']

Since quadrilateral ABCD is a parallelogram and AC = 12, BD = 16, AB = CD = 11, therefore OC = 0.5 × AC = 6, OD = 0.5 × BD = 8, therefore the perimeter of triangle DOC is: CD + OD + OC = 11 + 6 + 8 = 25.

The final answer is: 25


In ▱ABCD, E is the midpoint of BC. DE is perpendicular to AE. F is the midpoint of AD. Connect EF. AE = 6, DE = 8. The length of AB is () option:['3', '4', '5', '6']

Taking into consideration the question, the image is illustrated below:

Since DE is perpendicular to AE, we have ∠AED = 90°. According to the Pythagorean theorem, we have AD = √{AE²+DE²} = √{6²+8²} = 10. Since ABCD is a parallelogram, we have AD parallel to BC and AD = BC. Since E is the midpoint of BC and F is the midpoint of AD, we have AF = 0.5×AD and BE = 0.5×BC. Therefore, AF is parallel to BE and AF = BE. Thus, ABEF is a parallelogram, and AB = EF. Since ∠AED = 90° and E is the midpoint of AD, AD = 10. Therefore, EF = 0.5×AD = 5. Thus, AB = EF = 5.

The final answer is: 5


Quadrilateral ABCD is a rhombus, the coordinates of points A and B are (√3, 0) and (0, 1) respectively. Points C and D lie on the coordinate axes. Therefore, the perimeter of rhombus ABCD is () option:['8', '4', '6', '5']

Following the given question, here is the image we've created:

Since the coordinates of points A and B are (√{3}, 0) and (0, 1) respectively, we have OB = 1 and OA = √{3}. Therefore, AB = √{OB²+OA²} = √{1+3} = 2. Since quadrilateral ABCD is a diamond, AB = BC = CD = DA = 2. Therefore, the perimeter of diamond ABCD is equal to 4×2 = 8.

The final answer is: 8


In parallelogram ABCD, AD is equal to 8 and AB is equal to 3. AE bisects ∠BAD and intersects BC at point E. The length of CE is () option:['2', '3', '4', '5']

Since quadrilateral ABCD is a parallelogram, we have BC = AD = 8. AD is parallel to BC. Therefore, ∠DAE = ∠BEA. Since AE bisects ∠BAD, we have ∠BAE = ∠DAE. Therefore, ∠BEA = ∠BAE. Thus, BE = AB = 3. Therefore, CE = BC - BE = 8 - 3 = 5.

The final answer is: 5


In rectangle ABCD, AB = 12. Point E is a point on AD, and AE = 6. The perpendicular bisector of BE intersects the extension of BC at point F. Connecting EF intersects CD at point G. If G is the midpoint of CD, the length of BC is () option:['12.5', '12', '10', '10.5']

Since G is the midpoint of CD in rectangle ABCD and AB = 12, we have CG = DG = 0.5 × 12 = 6. In triangles DEG and CFG, ∠D = ∠DCFDG = CG and ∠DGE = ∠CGF. Therefore, triangles DEG and CFG are congruent by the ASA (Angle-Side-Angle) criterion. Thus, DE = CF and EG = FG. Assuming DE = x, we have BF = BC + CF = AD + CF = 6 + x + x = 6 + 2x. In right triangle DEG, EG = √(DE² + DG²) = √(x² + 6²). Therefore, EF = 2√(x² + 6²). Since FH is perpendicular and bisects BE, we have BF = EF. Therefore, 6 + 2x = 2√(x² + 6²). Solving for x, we have x = 4.5. Thus, AD = AE + DE = 6 + 4.5 = 10.5. Therefore, BC = AD = 10.5.

The final answer is: 10.5


Quadrilateral ABCD is a rhombus. AC is equal to 8cm, DB is equal to 6cm. DH is perpendicular to AB at H. Therefore, DH is equal to (). option:['3.6', '4.8', '5', '10']

Assuming AC and BD intersect at point O, since the quadrilateral ABCD is a rhombus with AC = 8cm and DB = 6 cm, we have OA = 0.5 × AC = 4 (cm) and OB = 0.5 × BD = 3 (cm). AC is perpendicular to BD, so ∠AOB = 90°. Therefore, AB = √(OA² + OB²) = √(4² + 3²) = 5 (cm). Since the area of the rhombus ABCD = AB * DH = 0.5 × AC * BD = 0.5 × 8 × 6 = 24 (cm²), DH = 24 / 5 = 4.8 (cm).

The final answer is: 4.8


In quadrilateral ▱ABCD, diagonals AC and BD intersect at point O, and AC equals 4. Points E, F, and G are the midpoints of AO, OB, and OC, respectively. Connect EF and FG. If the perimeter of △EFG is 7, then the perimeter of ▱ABCD is (). option:['10', '15', '20', '25']

As dictated by the question, the image is shown as follows:

Since quadrilateral ABCD is a parallelogram, we have OA = OC, AB = CD, and AD = BC. Since E, F, and G are the midpoints of AO, OB, and OC respectively, we have EG = 0.5 × AC, EF = 0.5 × AB, and FG = 0.5 × BC. Since AC = 4, we have EG = 2. Since the perimeter of triangle EFG is 7, we have EF + FG = 7 - 2 = 5. Thus, AB + BC = 2EF + 2FG = 2 × (EF + FG) = 2 × 5 = 10. Therefore, the perimeter of quadrilateral ABCD is 2AB + 2BC = 2 × 10 = 20.

The final answer is: 20


In the parallelogram ABCD, AC intersects BD at point O. ∠BAC is equal to 90°. Point E is the midpoint of BC and AE is equal to 3.5cm. The perimeter of triangle AOD is 3cm greater than the perimeter of triangle AOB. Therefore, the perimeter of parallelogram ABCD is () option:['20cm', '21cm', '22cm', '23cm']

Following the question, we present the image as follows:

Since quadrilateral ABCD is a parallelogram, we have OB = OD, AB = CD, and AD = BC. Since the perimeter of triangle AOD is 3 cm more than the perimeter of triangle AOB, we have (OA + OD + AD) - (OA + OB + AB) = AD - AB = 3 cm. Since ∠BAC = 90° and E is the midpoint of BC, we have AE = 0.5×BC. Therefore, BC = 2AE = 2×3.5 = 7 cm. Thus, AB = 7 - 3 = 4 cm. Therefore, the perimeter of parallelogram ABCD is 2(4+7) = 22 cm.

The final answer is: 22cm


Triangle ABC has a perimeter of 4, where points D, E, F are the midpoints of AB, BC, and CA, respectively. The perimeter of triangle DEF is () option:['1', '2', '3', '4']

Based on the given question, the picture is illustrated below:

Since the perimeter of triangle ABC is 4, we have AB + AC + BC = 4. Since points D, E, F are midpoints of AB, BC, CA, we have EF = 0.5 × AB, DE = 0.5 × AC, DF = 0.5 × BC. Therefore, the perimeter of triangle DEF = EF + DE + DF = 0.5 × (AB + AC + BC) = 2.

The final answer is: 2


In Rt△ABC, ∠ACB is equal to 90°. D, E, and F are the midpoints of sides AB, AC, and BC, respectively. Connect EF and CD. If EF is equal to 5 and BC is equal to 4, the length of CD is (). option:['5', '6', '8', '10']

In accordance with the provided context, the image is presented below:

Since triangle ABC is a right triangle, CD is the median of the hypotenuse. Therefore, CD = 0.5×AB. Since E and F are the midpoints of sides AC and BC, EF is the median of triangle ABC. Therefore, EF = 0.5×AB. Therefore, CD = EF = 5.

The final answer is: 5


In ▱ABCD, point F on BC bisects ∠BAD while point E on AD bisects ∠ABC. Connecting EF, if AF = 6 and BE = 8, then the length of AB is (). option:['6', '5', '4', '3']

Since quadrilateral ABCD is a parallelogram, we have AD ∥ BC and ∠BAD + ∠ABC = 180°. Therefore, line AF bisects ∠BAD and intersects BC at point F, while line BE bisects ∠ABC and intersects AD at point E. Thus, ∠BAF = 0.5×∠BAD and ∠ABE = 0.5×∠ABC. Consequently, ∠BAF + ∠ABE = 90°, therefore AF ⊥ BE. Given AF = 6 and BE = 8, we have AO = 3, BO = 4, and AB = √{3²+4²} = 5.

The final answer is: 5


In ▱ABCD, diagonal lines AC and BD intersect at point O. Point E is the midpoint of BC. Connecting OE, if OE=8, then the length of AB is () option:['16', '18', '20', '22']

∵ The quadrilateral ABCD is a parallelogram, and the diagonals AC and BD intersect at point O. Therefore, point O is the midpoint of AC and BD. Since point E is the midpoint of BC, then OE is the median of △ABC, meaning OE = 0.5×AB. Therefore, since OE = 8, AB = 16.

The final answer is: 16


In quadrilateral ABCD, diagonal AC intersects BD at point O. Point E is the midpoint of BC. If the perimeter of quadrilateral ABCD is 10 and OE is 1, the length of BC is (). option:['4', '3', '2', '1']

Since quadrilateral ABCD is a parallelogram, diagonals AC and BD intersect at point O. Therefore, O is the midpoint of AC and BC, AB = CD, and AD = BC. Since point E is the midpoint of BC, therefore OE = 0.5 × AB = 1, hence AB = CD = 2. Given that the perimeter of quadrilateral ABCD is 10, therefore 2BC = 10 - 2AB = 6, hence BC = 3.

The final answer is: 3


AD is the angle bisector of △ABC, and point E is the midpoint of AC. Connect DE. If AB=AC=10 and BC=8, then the perimeter of △CDE is (). option:['20', '12', '13', '14']

Adhering to the provided question, the picture is presented as follows:

Since AB = AC and AD is the angle bisector of triangle ABC, and BC = 8, therefore AD ⊥ BC. CD = BD = 0.5 × BC = 4. In right triangle ADC, point E is the midpoint of AC, therefore DE = CE = AE = 0.5 × AC = 5. AC = 10, therefore the perimeter of triangle CDE = CD + CE + DE = 14.

The final answer is: 14


In △ABC, D and E are the midpoints of AB and AC, respectively. F is a point on DE. AF and CF are connected and ∠AFC is equal to 90°. If DF is equal to 1 and AC is equal to 6, then the length of BC is (). option:['5', '6', '7', '8']

In right triangle AFC, point E is the midpoint of side AC, AC = 6. Therefore, EF = 0.5 × AC = 3. Thus, DE = DF + EF = 3 + 1 = 4. Since points D and E are the midpoints of sides AB and AC, we have BC = 2DE = 8.

The final answer is: 8


In ▱ABCD, connect AC. If ∠ABC = ∠CAD = 60° and AB = 3, then the length of AD is (). option:['3', '6', '9', '18']

Since quadrilateral ABCD is a parallelogram, AB = 3, ∠ABC = 60°. Therefore, ∠ABC = ∠ADC = 60°, CD = AB = 3, since ∠ABC = ∠CAD = 60°, therefore ∠ADC = ∠DAC = 60°, thus triangle ADC is an equilateral triangle, therefore AD = CD = 3.

The final answer is: 3


In quadrilateral ABCD, the perimeter is 36. Diagonals AC and BD intersect at point O. Point E is the midpoint of CD and BD = 12. Connect OE. The perimeter of triangle DOE is (). option:['15', '18', '21', '24']

According to the given question, the image is depicted as follows:

Since the perimeter of parallelogram ABCD is 36, we have BC + CD = 18. Since OD = OB and DE = EC, we have OE + DE = 0.5 × (BC + CD) = 9. Since BD = 12, OD = 0.5 × BD = 6. Therefore, the perimeter of triangle DOE is 9 + 6 = 15.

The final answer is: 15


In figure ▱ABCD, the diagonals AC and BD intersect at point O, and AC+BD=16. If the perimeter of △BCO is 14, then the length of BC is (). option:['12', '9', '8', '6']

Following the given question, here is the image we've created:

Since quadrilateral ABCD is a parallelogram, we have AO = CO, BO = DO, AB = CD = 6. Also, since AC + BD = 16, CO + BO = 8. The perimeter of triangle BCO is 14, thus OC + OB + BC = 14, and therefore BC = 6.

The final answer is: 6


In rectangle ABCD, point O is the midpoint of diagonal AC, and point M is the midpoint of AD. If AB = 5 and AD = 12, the length of OM + OB is (). option:['7', '8', '9', '10']

As per the question, the image is represented below:

- Thus, since the quadrilateral ABCD is a rectangle, we have ∠ABC = 90°, BC = AD = 12, CD = AB = 5. Therefore, AC = √{AB²+BC²}=√{5²+12²}=13.
- Since O is the midpoint of the diagonal AC of rectangle ABCD, and M is the midpoint of AD, we have OB = 0.5×AC = 6.5.
- Additionally, OM is the median of triangle ACD, so OM = 0.5×CD = 2.5.
- Therefore, OM + OB = 2.5 + 6.5 = 9.

The final answer is: 9


In △ABC, points D and E are respectively the midpoints of the sides AB and BC. Connecting DE, if the perimeter of △BDE is 6, then the perimeter of △ABC is () option:['8', '10', '12', '14']

As dictated by the question, the image is shown as follows:

Since the perimeter of triangle BDE is 6, we have BD + BE + DE = 6. Since D and E are the midpoints of the sides AB and BC, we have AB = 2BD, BC = 2BE, AC = 2DE. Therefore, the perimeter of triangle ABC = AB + BC + AC = 2(BD + BE + DE) = 2*6 = 12.

The final answer is: 12


In rectangle ABCD, the two diagonals intersect at point O. ∠AOB is equal to 60° and AC is equal to 8. The length of side AB is (). option:['4', '5', '6', '8']

Taking into consideration the question, the image is illustrated below:

Since quadrilateral ABCD is a rectangle, we have AO = 0.5×AC = 0.5×8 = 4, AC = BD, and OB = 0.5×BD. Thus, AO = OB. Since ∠AOB = 60°, triangle AOB is an equilateral triangle, so AB = AO = 4.

The final answer is: 4


In triangle △ABC, points D and E are the midpoints of sides AB and AC, respectively. If DE = 4, then BC equals () option:['2', '4', '8', '10']

Following the question, we present the image as follows:

Since D and E are the midpoints of AB and AC, respectively, we have DE = 4. Therefore, BC = 2DE = 2 × 4 = 8.

The final answer is: 8


In △ABC, points D, E, and F are the midpoints of the three sides. AB = 6, AC = 4. Connect DE and DF, then the perimeter of quadrilateral AEDF is () option:['40', '30', '20', '10']

Since in △ABC, E, D, and F are the midpoints of AB, BC, and CA, we have DE = AF = 0.5×AC = 2, DF = AE = 0.5×AB = 3. Therefore, the perimeter of quadrilateral AEDF is (2+3)×2 = 10.

The final answer is: 10


In rhombus ABCD with a side length of 10, E is the midpoint of AD, and O is the intersection of the diagonals. One side FG of rectangle OEFG lies on AB, and EF = 4. The length of BG is () option:['3', '2', '√{2}', '1']

In accordance with the provided context, the image is presented below:

Since the quadrilateral ABCD is a rhombus, we have BD ⊥ AC and AB = AD = 10. Thus, ∠AOD = 90°. Since E is the midpoint of AD, we have OE = AE = 0.5×AD = 5. Since the quadrilateral OEFG is a rectangle, we have FG = OE = 5. Since AE = 5 and EF = 4, we have AF = √{AE²-EF²} = √{5²-4²} = 3. Therefore, BG = AB - AF - FG = 10 - 3 - 5 = 2.

The final answer is: 2


In ▱ABCD, AD = 7, AB = 5, DE bisects ∠ADC and intersects BC at point E. The length of BE is () option:['2', '3', '4', '5']

Since quadrilateral ABCD is a parallelogram, we have BC = AD = 7, CD = AB = 5, and AD is parallel to BC. Therefore, ∠ADE = ∠DEC. Since DE bisects ∠ADC, we have ∠ADE = ∠CDE, thus ∠CDE = ∠DEC. Therefore, EC = CD = 5, so BE = BC - EC = 2.

The final answer is: 2


In rectangle ABCD, the diagonals AC and BD intersect at point O. Points E and F are the midpoints of AB and AO, respectively, and AC = 8. Connect EF, then the length of EF is (). option:['2', '4', '6', '8']

Since quadrilateral ABCD is a rectangle, we have AC = BD = 8. Also, BO = DO = 0.5×BD, so BO = DO = 0.5×BD = 4. Additionally, points E and F are the midpoints of AB and AO, respectively. Therefore, EF is the median of triangle AOB. Hence, EF = 0.5×BO = 2.

The final answer is: 2


In ∆ABC, points D, E, and F are the midpoints of sides AB, BC, and CA, respectively, and AB = 8cm, AC = 10cm. If we connect DE and EF, then the perimeter of quadrilateral ADEF is equal to () option:['14', '18', '20', '24']

According to the given question, the image is depicted as follows:

Since points D, E, and F are the midpoints of sides AB, BC, and CB, and AB = 8cm, AC = 10cm, we have AD = 0.5×AB = 4cm, DE = 0.5×AC = 5cm, AF = 0.5×AC = 5cm, EF = 0.5×AB = 4cm. Therefore, the perimeter of quadrilateral ADEF = AD+DE+EF+AF = 18cm.

The final answer is: 18


The perimeter of ▱ABCD is 40, AD:AB=3:2, then the length of BC is () option:['8', '12', '16', '24']

Since the perimeter of square ABCD is 40, and AD:AB = 3:2, if we let AD be 3x and AB be 2x, we can get: 3x+2x = 20, and solving for x gives: x = 4. Therefore, AD = BC = 12.

The final answer is: 12


In ▱ABCD, if the perimeter of △ACD is 13cm and AC = 4cm, the perimeter of ▱ABCD is () option:['17', '', '18', '', '22', '', '34', '']

Since AC = 4cm, the perimeter of triangle ADC is 13cm. Therefore, AD + DC = 13 - 4 = 9 cm. Furthermore, since quadrilateral ABCD is a parallelogram, we have AB = CD and AD = BC. Therefore, the perimeter of the parallelogram is 2(AB + BC) = 18cm.


In rhombus ABCD, diagonal AC intersects diagonal BD at point O. Point H is the midpoint of side AD. The perimeter of the rhombus is 28. If OH is connected, then the length of OH is (). option:['3.5', '4.5', '7', '14']

Since the perimeter of rhombus ABCD is 28, we have AC ⊥ BD, AB = BC = CD = AD = 7. Therefore, ∠AOD = 90°. Since H is the midpoint of side AD, we have OH = 0.5×AD = 3.5.

The final answer is: 3.5


In ▱ABCD, AC = 5cm. If the perimeter of △ACD is 14cm, then the perimeter of ▱ABCD is () option:['18cm', '19cm', '28cm', '38cm']

- ∵ The perimeter of △ACD is 14cm, which means AD+CD+AC = 14cm, and AC = 5cm. Therefore, AD+CD = 9cm. Since quadrilateral ABCD is a parallelogram, AB = CD and BC = AD, then the perimeter of ▱ABCD is AB+BC+AD+CD = 9+9 = 18 (cm).

The final answer is: 18cm


Given that diamond ABCD, connecting AC and BD, BD = 8 and the area is 24, the perimeter of diamond ABCD is () option:['5', '10', '10√{2}', '20']

Assume AC intersects BD at point O. Since quadrilateral ABCD is a rhombus, we have BD = 8. Therefore, AB = BC = CD = AD and OB = 0.5×BD = 4. We also have OA = OC, and AC ⊥ BD. Since the area of rhombus ABCD is 24, we have 0.5×AC×BD = 24, which means 0.5×AC×8 = 24. Therefore, AC = 6, and OA = 3. In right triangle AOB, according to the Pythagorean theorem, we have AB = √(OA²+OB²) = √(3²+4²) = 5. Thus, the perimeter of rhombus ABCD is 4AB = 20.

The final answer is: 20


In △ABC, AB is equal to 7, AC is equal to 6, and BC is equal to 5. D and E are the midpoints of AB and AC, respectively. Connect DE. The length of DE is (). option:['4', '3.5', '3', '2.5']

As dictated by the question, the image is shown as follows:

Since D and E are the midpoints of AB and AC, we have DE as the median of △ABC, so DE = 0.5×BC = 2.5.

The final answer is: 2.5


In rhombus ABCD, diagonal AC intersects BD at point O, while E is the midpoint of DC. If the perimeter of the rhombus is 16 and the length of OE is (), then the answer is (). option:['2', '1', '4', '3']

Abiding by the question, the image is depicted below:

Since quadrilateral ABCD is a rhombus, we have AB = BC = CD = AD and BO = DO. As the perimeter of rhombus ABCD is 16, BC = 4. Since point E is the midpoint of CD, OE is the median of triangle BCD, therefore OE = 0.5 × BC = 2.

The final answer is: 2


In ▱ABCD, diagonal AC intersects diagonal BD at point O. If AD = 5, AC = 10, and BD = 6, the perimeter of △BOC is (). option:['13', '16', '18', '21']

Following the question, we present the image as follows:

Since the two diagonals of square ABCD intersect at point O and AC = 10, BD = 6, AD = 5, we have BO = DO = 3, AO = CO = 5, and BC = AD = 5. Therefore, the perimeter of triangle BOC is: BO + CO + BC = 3 + 5 + 3 = 13.

The final answer is: 13


In △ABC, point M is the midpoint of BC. AD bisects ∠BAC, BD is perpendicular to AD at point D, and extended to intersect AC at point N. If AB = 10 and AC = 16, then the length of MD is (). option:['3', '4', '6', '7.5']

Adhering to the provided question, the picture is presented as follows:

Since AD bisects ∠BAC, we have ∠BAD = ∠NAD. Since BD is perpendicular to AD at point D, we have ∠ADB = ∠ADN = 90°. In triangle ADB and ADN, we have ∠BAD = ∠NAD, AD = AD, and ∠ADB = ∠ADN. Therefore, triangle ADB is congruent to triangle ADN by ASA congruence. Hence, AN = AB = 10 and BD = DN. Thus, NC = AC - AN = 16 - 10 = 6. Since M is the midpoint of BC, we have BM = CM. Since BD = DN and BM = MC, we have DM = 0.5×NC = 3.

The final answer is: 3


In △ABC, D, E and F are the midpoints of BC, AC and AB, respectively. Point H is the foot of the altitude from A to BC. Connecting FD, EF and HE, if FD = 8, then HE is equal to () option:['4', '6', '8', '10']

Following the given question, here is the image we've created:

∵D and F are respectively the midpoints of AB and BC, therefore DF is the median of △ABC. Thus, DF = 0.5×AC (Median theorem of triangle). Also, since E is the midpoint of segment AC, AH is perpendicular to BC, therefore EH = 0.5×AC, so EH = DF = 8.

The final answer is: 8


In rectangle ABCD, diagonal AC intersects diagonal BD at point O. DE is perpendicular to AC, and its foot is E. If AO:OE is 3:2 and DE is equal to 2√{5}, then the length of CE is (). option:['1', '2', '√{5}', '√{2}']

According to the given question, the image is depicted as follows:

Since quadrilateral ABCD is a rectangle, we have AC = BD, OD = 0.5×BD, OC = 0.5×AC, ∠ADC = 90°. Therefore, OC = OD. Since AO:OE = 3:2, DE ⊥ AC, let OA = 3x and OE = 2x. In right triangle DOE, OD² = OE² + DE², that is (3x)² = (2x)² = (2√{5})². Solving, x = 2 (negative 2 is discarded), thus OC = OA = 6, OE = 4, therefore EC = OC - OE = 6 - 4 = 2.

The final answer is: 2


In ▱ABCD, BE bisects ∠ABC and intersects AD at point E. AE is equal to 2 and ED is equal to 1. The perimeter of ▱ABCD is (). option:['14', '12', '10', '8']

Based on the given question, the picture is illustrated below:

Since quadrilateral ABCD is a parallelogram, we have AD ∥ BC, AB = CD, and AD = BC. Therefore, ∠AEB = ∠EBC. Since BE bisects ∠ABC, we have ∠ABE = ∠EBC, which implies ∠ABE = ∠AEB. Therefore, AB = AE = 2, DE = 1, and AD = BC = 3. Thus, the perimeter of parallelogram ABCD is 2 × (3 + 2) = 10.

The final answer is: 10


In ▱ABCD, AE bisects ∠DAB and intersects CD at point E. If AB = 6 and CE = 2, the perimeter of ▱ABCD is (). option:['14', '16', '20', '24']

Since AE bisects ∠BAD, we have ∠DAE = ∠EAB. Also, since quadrilateral ABCD is a parallelogram, DC || AB. Therefore, ∠DEA = ∠EAB = ∠DAE. This implies DA = DE. Given AB = 6, CE = 2, we can deduce DE = AB - CE = 6 - 2 = 4. Therefore, the perimeter of parallelogram ABCD is 2(DA + AB) = 2*(4+6) = 20.

The final answer is: 20


The diagonal AC of quadrilateral ABCD intersects BD at point O. Point E is the midpoint of AB. Connecting OE, the perimeter of triangle BEO is 20, then the perimeter of triangle BCD is () option:['30', '40', '50', '60']

Taking into consideration the question, the image is illustrated below:

Since the diagonals AC and BD of ▱ABCD intersect at point O, we have BO = DO = 0.5 × BD, BD = 2OB, thus O is the midpoint of BD. Since point E is the midpoint of AB, AB = 2BE, BC = 2OE. Since the quadrilateral ABCD is a parallelogram, AB = CD, thus CD = 2BE. Since the perimeter of △BEO is 20, OB + OE + BE = 20, therefore BD + BC + CD = 2OB + 2OE + 2BE = 2(OB + OE + BE) = 40, thus the perimeter of △BCD is 40.

The final answer is: 40


In quadrilateral ABCD, CE bisects angle DCB, intersects AB at point E, AE=2, DC=6, the perimeter of quadrilateral ABCD is () option:['16', '18', '20', '24']

As per the question, the image is represented below:

In quadrilateral ABCD, CE bisects ∠BCD, so ∠DCE = ∠ECB. AB is parallel to DC and AB = DC, so ∠BEC = ∠DCE. Therefore, ∠BEC = ∠BCE, which means BE = BC = AD. Since AB = DC = 6 and AE = 2, we have BE = BC = AD = 4. Thus, the perimeter of parallelogram ABCD is 2(4+6) = 20.

The final answer is: 20


In right triangle △ABC, ∠ACB=90°, points G, E, D are the midpoints of sides AB, BC, CA, respectively. Connecting DE, CG, if DE+CG=7, then the length of CG is () option:['3', '3.5', '4', '5']

As dictated by the question, the image is shown as follows:

In right triangle ABC, ∠ACB = 90°. Point G is the midpoint of side AB, therefore CG = 0.5×AB. Since points E and D are the midpoints of sides BC and CA respectively, therefore DE is the median of △ABC, thus DE = 0.5×AB, and DE = CG. Since DE + CG = 7, therefore CG = DE = 3.5.

The final answer is: 3.5


In ▱ABCD, AE is perpendicular to BC at point E, and AF is perpendicular to CD at point F, with AE = 3 and AF = 4. If the perimeter of ▱ABCD is 56, then the length of BC is () option:['14', '16', '28', '32']

Since the perimeter of square ABCD is 56, we have BC + CD = 28. In square ABCD, AE ⊥ BC and AF ⊥ CD. Therefore, the area of square ABCD is equal to BC • AE = CD • AF. Given AE = 3, AF = 4, we have 3BC = 4CD. Thus, BC = 16, CD = 12.

The final answer is: 16


AC is the diagonal of rectangle ABCD. The bisector of ∠BAC intersects BC at point E. BM is perpendicular to AE at point M, and intersects AC at point F. If N is the midpoint of BC, connect MN. Given AB = 6 and BC = 8, the length of MN is (). option:['3.5', '3', '2.5', '2']

∵Quadrilateral ABCD is a rectangle, therefore ∠ABC = 90°. Since AB = 6 and BC = 8, we have AC = √{AB²+BC²}=√{6²+8²}=10. Since the angle bisector of ∠BAC intersects BC at point E, and BM ⊥ AE at point M, therefore triangle ABF is an isosceles triangle, thus BM = MF and AB = AF, therefore FC = AC - AF = AC - AB = 10 - 6 = 4. Since point N is the midpoint of BC, therefore MN is the median of triangle BFC, hence 2MN = FC = 4, thus MN = 2.

The final answer is: 2


The perimeter of rhombus ABCD is 16. Connecting AC and BD, ∠DAB = 60°, then the length of BD is () option:['10', '8', '4', '2']

Since the perimeter of rhombus ABCD is 16, we have AB = AD = BC = CD = 4. Since ∠DAB = 60°, triangle ADB is an equilateral triangle. Therefore, BD = AB = 4.

The final answer is: 4


In the right triangle △ABC as shown in the figure, ∠C is equal to 90° and AB is equal to 10cm. Point D is the midpoint of AB. The length of CD is (). option:['7cm', '6cm', '5cm', '4cm']

In accordance with the provided context, the image is presented below:

Since ∠C = 90°, AB = 10cm, and point D is the midpoint of AB, we have CD = 0.5×AB = 0.5×10 = 5 (cm).

The final answer is: 5cm


The diagonal AC of the parallelogram intersects with the diagonal BD at point O. Given that BC = 6, BD = 12, and AC = 8, then the perimeter of triangle OAD is () option:['13', '14', '16', '18']

Since quadrilateral ABCD is a parallelogram, we have OA = OC = 4, OB = OD = 6, and BC = AD = 6. Therefore, the perimeter of triangle OAD is equal to OD + OA + AD = 4 + 6 + 6 = 16.

The final answer is: 16


In right triangle △ABC, ∠ACB is equal to 90°. Points D and E are the midpoints of sides AB and AC, respectively. BC is extended to point F such that CF is equal to half of BC. CD, DE, and EF are connected. If EF is equal to 5, then the length of side AB is () option:['10', '12', '15', '18']

Since points D and E are the midpoints of sides AB and AC, we have DE = 0.5 × BC and DE parallel to BC. Moreover, CF = 0.5 × BC, so DE = CF and DE parallel to CF. Therefore, quadrilateral DEFC is a parallelogram, and thus CD = EF = 5. Since ∠ACB = 90° and point D is the midpoint of side AB, we have AB = 2CD = 10.

The final answer is: 10


Rectangle ABCD's diagonal AC is equal to 8. AC intersects BD at point O. ∠BOC is equal to 120°. The length of AB is () option:['3', '4', '5', '6']

∵Quadrilateral ABCD is a rectangle, therefore AC = BD, AO = BO = CO = DO = 4. Since ∠BOC = 90°, therefore ∠AOB = 60°, thus △BOA is an equilateral triangle, therefore AB = AO = 4.

The final answer is: 4


In △ABC, points D and E are the midpoints of AB and AC, respectively. Point F is on DE, and AF is perpendicular to CF. If AC = 3 and BC = 6, then the length of DF is () option:['1.5', '1', '0.5', '2']

Since D and E are the midpoints of AB and AC, BC = 6, we have DE = 0.5 x BC = 3. Since AF is perpendicular to CF, we have ∠AFC = 90°. Since E is the midpoint of AC, AC = 3, thus FE = 0.5 x AC = 1.5, therefore DF = DE - FE = 1.5.

The final answer is: 1.5


In ▱ABCD, DE bisects ∠ADC, intersecting BC at point E, AD = 9, BE = 3, then the perimeter of ▱ABCD is () option:['15', '24', '30', '36']

Based on the given question, the picture is illustrated below:

In quadrilateral ABCD, AD = 9, so BC = AD = 9. Since AD is parallel to BC, we have CE = BC - BE = 9 - 3 = 6. ∠ADE = ∠CED and DE bisects ∠ADC. Hence, ∠ADE = ∠CDE and ∠CDE = ∠CED. Therefore, CD = CE = 6. The perimeter of quadrilateral ABCD is: 2(AD + CD) = 30.

The final answer is: 30


In rhombus ABCD, points A, C, and B, D are connected at point O. AB is equal to 13 and AC is equal to 10. The length of BD is () option:['24', '16', '12', '8']

Since quadrilateral ABCD is a rhombus, we have OA = OC = 0.5×AC = 5. Also, OB = OD and AC ⊥ BD. Thus, ∠AOB = 90°. Consequently, OB = √{AB²-OA²} = √{13²-5²} = 12. Therefore, BD = 2OB = 24.

The final answer is: 24


In parallelogram ABCD, diagonals AC and BD intersect at point O. E is the midpoint of AB. Connect OE. If AE + EO = 4, then the perimeter of parallelogram ABCD is (). option:['16', '20', '12', '8']

Taking into consideration the question, the image is illustrated below:

Since quadrilateral ABCD is a parallelogram, we have OA = OC. Also, since AE = EB, then OE = 0.5 × BC. Given that AE + EO = 4, we have 2AE + 2EO = 8. Therefore, AB + BC = 8, and hence the perimeter of parallelogram ABCD is 2 × 8 = 16.

The final answer is: 16


In Rt△ABC, D is the midpoint of AB, AB = 10, then CD is () option:['3', '4', '5', '6']

According to the given question, the image is depicted as follows:

Since ∠ACB = 90° and D is the midpoint of AB, we have CD = 0.5 × AB. Since AB = 10, CD = 5.

The final answer is: 5


In Rt△ABC, ∠C is equal to 90°. AB is equal to 10cm and AC is equal to 8cm. Point D is a point on AC. Using AD and BD as adjacent sides, we construct a parallelogram ADBE. The minimum value of the diagonal DE is (). option:['4', '6', '8', '10']

-AB intersects DE at point O, since in right triangle ABC, ∠B = 90°, we have BC ⊥ AC. Since quadrilateral ADBE is a parallelogram, we have OD = OE and OA = OB. Therefore, when OD is minimized, segment DE is shortest, at which point OD ⊥ AC. Thus, OD ∥ CB. Also, point O is the midpoint of AB, so OD is the median of △ABC, and OD = 0.5 × CB. Since ∠C = 90°, AB = 10cm, AC = 8cm, so BC = √(AB²-AC²) = √(10²-8²) = 6 (cm), thus OD = 3cm, and ED = 2OD = 6.

The final answer is: 6


In rhombus ABCD, points E and F are the midpoints of AD and AC, respectively. If EF = 3, what is the perimeter of rhombus ABCD? () option:['6', '18', '24', '30']

Abiding by the question, the image is depicted below:

Since E and F are the midpoints of AD and AC respectively, we have CD = 2EF = 6. Since quadrilateral ABCD is a rhombus, we have AB = BC = CD = AD = 6. Therefore, the perimeter of rhombus ABCD = 4 * 6 = 24.

The final answer is: 24


In ▱ABCD, points A, C, and BD intersect at point O. Point E is the midpoint of AD, and OE is connected. If OE = 2, the length of CD is (). option:['1', '2', '3', '4']

Adhering to the provided question, the picture is presented as follows:

Since quadrilateral ABCD is a parallelogram, we have AO = CO. Since point E is the midpoint of side CD, we have EO = 0.5 × CD. Therefore, CD = 2OE = 4.

The final answer is: 4


In quadrilateral ABCD, the angle bisector of ∠BCD intersects the extension of BA at point E, and AE = 2, AD = 5. The length of CD is (). option:['4', '3', '2', '1.5']

In accordance with the provided context, the image is presented below:

Since quadrilateral ABCD is a parallelogram, we have AB || CD, AD = BC = 5, and CD = AB. Therefore, ∠E = ∠ECD. Since CE bisects ∠BCD, we have ∠BCE = ∠ECD. Therefore, ∠E = ∠BCE, which implies BE = BC = 5. Hence, AB = BE - AE = 5 - 2 = 3. Thus, CD = 3.

The final answer is: 3


In rhombus ABCD, diagonals AC and BD intersect at point O. E is the midpoint of CD. Connecting OE, if OE = 3, then the length of AB is () option:['6', '5', '4', '3']

As per the question, the image is represented below:

Since quadrilateral ABCD is a rhombus, we have AC ⊥ BD and CD = AB. Since point E is the midpoint of CD, we have CD = 2OE = 6. Therefore, AB = 6.

The final answer is: 6


In parallelogram ABCD, diagonal AC intersects at point O, and AD > CD. Perpendicular line OM is drawn from point O to AC, intersecting AD at point M. If the perimeter of △CDM is 7.5, then the perimeter of parallelogram ABCD is (). option:['7.5', '15', '17', '19']

∵ Quadrilateral ABCD is a parallelogram, ∴ AO = CO, ∵ OM ⊥ AC, ∴ AM = MC, ∵ the perimeter of triangle CDM is 7.5, ∴ CM + DM + CD = 7.5 = AM + DM + CD, ∴ AD + CD = 7.5, ∴ the perimeter of parallelogram ABCD = 2×7.5 = 15.

The final answer is: 15


In △ABC, AB=AC=10, BC=8, AD is perpendicular to BC at point D. Point E is the midpoint of AC. Connect DE. The length of DE is (). option:['4', '5', '6', '8']

Since AB = AC, AD ⊥ BC, therefore CD = DB, since point E is the midpoint of AC, therefore DE is the median of △ABC, therefore DE = 0.5×AB = 0.5×10 = 5.

The final answer is: 5


In parallelogram ABCD, diagonals AC and BD intersect at point O. E is the midpoint of side BC. AB = 4. If OE is connected, then OE = () option:['1', '2', '4', '8']

Based on the given question, the picture is illustrated below:

∵The quadrilateral ABCD is a parallelogram, therefore OA = OC. Also, since point E is the midpoint of BC, OE is the median of △ABC. Therefore, according to the theorem of medians in a triangle, we can obtain: AB = 2OE = 4. Then, OE = 2.

The final answer is: 2


In △ABC, AB = 8, BC = 12, points D and E are the midpoints of sides AB and AC, respectively. Point F is a point on segment DE. Connect AF and BF, if ∠AFB = 90°, the length of segment EF is () option:['4', '3', '2', '1']

Since points D and E are the midpoints of sides AB and AC, we have DE as the median of △ABC. Given that BC = 12, then DE = 0.5 × BC = 0.5 × 12 = 6. Since ∠AFB = 90°, D is the midpoint of AB, AB = 8, thus DF = 0.5 × AB = 0.5 × 8 = 4, and EF = DE - DF = 6 - 4 = 2.

The final answer is: 2


The distance from the midpoint M of the side of rhombus ABCD to the intersection point O of the diagonals is 3cm, then the perimeter of rhombus ABCD is () option:['10cm', '12cm', '16', '', '24', '']

As dictated by the question, the image is shown as follows:

Since quadrilateral ABCD is a rhombus, we have AB = AD = CD = BC and BO = DO. Also, since point M is the midpoint of AB, we have AD = 2OM = 6cm. Therefore, the perimeter of rhombus ABCD is 4 × 6 = 24cm.


In △ABC, points D and E are the midpoints of AB and AC, respectively. Point F is a point on DE, connecting AF and CF, with ∠AFC = 90°. If BC = 12 and AC = 8, then the length of DF is () option:['1', '2', '3', '4']

Since point D and point E are the respective midpoints of AB and AC, we have DE as the median of triangle ABC. Thus, DE = 0.5×BC, and since BC = 12, DE = 6. In right-angled triangle AFC where ∠AFC = 90° and point E is the midpoint of AC, AC = 8. Therefore, FE = 0.5×AC = 4, and consequently, DF = DE - FE = 6 - 4 = 2.

The final answer is: 2


In the diamond ABCD, diagonals AC and BD intersect at point O. Point M is the midpoint of DC. If the perimeter of the diamond ABCD is 24, when connecting OM, the length of OM is (). option:['12', '8', '6', '3']

According to the given question, the image is depicted as follows:

Since the perimeter of rhombus ABCD is 24, we have DC = \frac{24}{4} = 6. Since quadrilateral ABCD is a rhombus, AC ⊥ BD. Since M is the midpoint of DC, in right triangle DOC, OM is the median on the hypotenuse. Therefore, OM = 0.5×DC = 3.

The final answer is: 3


In ▱ABCD, BD is perpendicular to AD. AB = 10 and AD = 6. A rectangle DEBF is constructed, where E and F are located on AB and CD, respectively. The length of the diagonal EF is (). option:['8', '9', '10', '11']

∵Since BD is perpendicular to AD, and AB = 10, AD = 6, therefore DB = 8. ∵DEBF is a rectangle, thus EF = DB = 8.

The final answer is: 8


In quadrilateral ABCD, points AC and BD intersect at point O. Given that ∠ADB = 90°, AC = 10, and AD = 4. The length of BD is () option:['4', '5', '6', '7']

Adhering to the provided question, the picture is presented as follows:

Since quadrilateral ABCD is a parallelogram, we have AO = CO = 0.5 × AC = 5, and BO = DO = 0.5 × BD. In right triangle ADO, DO = √{AO² - AD²} = √{5² - 4²} = 3. Therefore, BD = 6.

The final answer is: 6


In right triangle △ABC, ∠ACB = 90°. D is the midpoint of side AB, AB = 6. If CD is connected, then the length of CD is () option:['2', '3', '4', '6']

In Rt△ABC, since ∠ACB = 90° and D is the midpoint of side AB, we have CD = 0.5×AB. Since AB = 6, we can conclude that CD = 0.5×AB = 3.

The final answer is: 3


In △ABC, points D and E are the midpoints of BC and BA, respectively. If AC=4, then DE=() option:['1', '2', '4', '8']

Following the given question, here is the image we've created:

Since point D and point E are respectively the midpoints of BC and BA, we have DE as the median line of triangle ABC. Therefore, DE = 0.5×AC. Since AC = 4, we have DE = 0.5×4 = 2.

The final answer is: 2


How many people are there in the image? option:[16, 26, 36, 46]

To start off, let's eliminate the faces on the left side of the image.

The above illustration illustrates our discovery of a total of 10 people.In light of this, we continue the search for faces to the right in the image, then deleting the faces located.

Based on the visual representation above, we've pinpointed a count of 10 people.Based on this premise, we advance our examination of faces to the right in the image, eliminating the faces identified.

As shown in the above image, we have identified a total of 10 people.Given this context, we move on to the right in the image, searching for faces and eliminating them.

In the depicted image, our findings indicate the presence of 6 individuals.In overview, the image showcases a grand total of 36 individuals.

The final answer is: 36


How many people do you see in the image? option:[55, 65, 75, 85]

Our initial action involves removing the faces on the left side of the image.

The image above reveals our identification of 10 individuals.Leveraging this insight, we move forward with our investigation of faces to the right side of the image, and remove the faces discovered.

As demonstrated above, our observation reveals the existence of 10 people.In light of this, we continue the search for faces to the right in the image, then deleting the faces located.

According to the illustration above, we've discovered a group of 10 individuals.Taking this into consideration, we extend our examination of faces to the right in the image, deleting them then.

The visual reference above confirms our identification of 10 individuals.Given this, we extend our search for faces to the right in the image, and delete the faces that are found.

As shown in the above image, we have identified a total of 5 people.To conclude, the total number of people in the image amounts to 55.

The final answer is: 55


Can you count the number of people present in the image? option:[21, 31, 41, 51]

Initially, we will remove the faces on the left side of the image.

Our analysis of the above image indicates the presence of 10 individuals.Given this context, we move on to the right in the image, searching for faces and eliminating them.

Based on the visual representation above, we've pinpointed a count of 10 people.Leveraging this insight, we move forward with our investigation of faces to the right side of the image, and remove the faces discovered.

The above illustration illustrates our discovery of a total of 10 people.Based on this premise, we advance our examination of faces to the right in the image, eliminating the faces identified.

The above illustration illustrates our discovery of a total of 1 people.Taking everything into consideration, the image contains a collective count of 31 people.

The final answer is: 31


Can you ascertain the quantity of people captured in the image? option:[41, 51, 61, 71]

To begin with, our first step will be to erase the faces located on the left side of the image.

The image above reveals our identification of 10 individuals.Building upon this, we proceed to the right in the image, searching for faces and removing them then.

In the depicted image, our findings indicate the presence of 10 individuals.Considering this development, we extend our search for faces to the right side of the image, and delete the faces found.

In the image above, we've successfully located 10 people.Given this context, we move on to the right in the image, searching for faces and eliminating them.

The image above reveals our identification of 10 individuals.Considering this development, we extend our search for faces to the right side of the image, and delete the faces found.

Our analysis of the above image indicates the presence of 1 individuals.Taking everything into consideration, the image contains a collective count of 51 people.

The final answer is: 51


Can you count the number of people present in the image? option:[15, 25, 35, 45]

Our initial action involves removing the faces on the left side of the image.

According to the illustration above, we've discovered a group of 5 individuals.In overview, the image showcases a grand total of 25 individuals.

The final answer is: 25


The visual reference above confirms our identification of 10 individuals.Considering this development, we extend our search for faces to the right side of the image, and delete the faces found.

As demonstrated above, our observation reveals the existence of 10 people.Given this context, we move on to the right in the image, searching for faces and eliminating them.

The visual reference above confirms our identification of 10 individuals.In light of this, we continue the search for faces to the right in the image, then deleting the faces located.

As demonstrated above, our observation reveals the existence of 1 people.To conclude, the total number of people in the image amounts to 31.

The final answer is: 31


How many people do you see in the image? option:[5, 15, 25, 35]

As a starting point, we'll initiate the process by eliminating the faces positioned on the left side of the image.

The above illustration illustrates our discovery of a total of 10 people.Given this, we extend our search for faces to the right in the image, and delete the faces that are found.

In the depicted image, our findings indicate the presence of 10 individuals.Based on this premise, we advance our examination of faces to the right in the image, eliminating the faces identified.

Based on the visual representation above, we've pinpointed a count of 5 people.In overview, the image showcases a grand total of 35 individuals.

The final answer is: 35


Can you count the number of people appearing in the image? option:[58, 68, 78, 88]

We initiate the process by deleting the faces on the left of the image.

In the image above, we've successfully located 10 people.Based on this premise, we advance our examination of faces to the right in the image, eliminating the faces identified.

In the image above, we've successfully located 10 people.Given this, we extend our search for faces to the right in the image, and delete the faces that are found.

In the image above, we've successfully located 10 people.Taking this into consideration, we extend our examination of faces to the right in the image, deleting them then.

In the image above, we've successfully located 10 people.Given this context, we move on to the right in the image, searching for faces and eliminating them.

Our analysis of the above image indicates the presence of 10 individuals.In light of this, we continue the search for faces to the right in the image, then deleting the faces located.

According to the illustration above, we've discovered a group of 8 individuals.Putting it all together, the comprehensive count of individuals in the image is 58.

The final answer is: 58


How many people are visible in the image? option:[39, 49, 59, 69]

Our first step involves deleting the faces visible on the left side of the image.

The visual reference above confirms our identification of 10 individuals.Building upon this, we proceed to the right in the image, searching for faces and removing them then.

Based on the visual representation above, we've pinpointed a count of 10 people.Leveraging this insight, we move forward with our investigation of faces to the right side of the image, and remove the faces discovered.

As shown in the above image, we have identified a total of 10 people.Building upon this, we proceed to the right in the image, searching for faces and removing them then.

The image above reveals our identification of 9 individuals.In summary, there are a total of 39 individuals in the image.

The final answer is: 39


Can you count the number of people appearing in the image? option:[33, 3, 13, 23]

According to the illustration above, we've discovered a group of 10 individuals.With this information in mind, we persist in our quest for faces to the right in the image, eliminating the faces discovered then.

Based on the visual representation above, we've pinpointed a count of 10 people.Expanding on this, we proceed to the right in the image, seeking faces and removing them.

The above illustration illustrates our discovery of a total of 3 people.Summing up, there are a combined 23 persons depicted in the image.

The final answer is: 23


Would you be able to figure out how many people are in the image? option:[18, 28, 38, 48]

We'll commence by deleting the faces on the left of the image.

In the image above, we've successfully located 10 people.In light of this, we continue the search for faces to the right in the image, then deleting the faces located.

Our analysis of the above image indicates the presence of 10 individuals.Taking this into consideration, we extend our examination of faces to the right in the image, deleting them then.

As shown in the above image, we have identified a total of 10 people.Expanding on this, we proceed to the right in the image, seeking faces and removing them.

The image above reveals our identification of 8 individuals.In conclusion, there are 38 people in total captured in the image.

The final answer is: 38


Our initial action involves removing the faces on the left side of the image.

The visual reference above confirms our identification of 10 individuals.Expanding on this, we proceed to the right in the image, seeking faces and removing them.

As demonstrated above, our observation reveals the existence of 10 people.Based on this premise, we advance our examination of faces to the right in the image, eliminating the faces identified.

In the image above, we've successfully located 10 people.Given this, we extend our search for faces to the right in the image, and delete the faces that are found.

Based on the visual representation above, we've pinpointed a count of 1 people.Summing up, there are a combined 41 persons depicted in the image.

The final answer is: 41


Could you provide a count of the people in the given image? option:[7, 17, 27, 37]

Initially, our focus will be on removing the faces located on the left side of the image.

As demonstrated above, our observation reveals the existence of 10 people.Leveraging this insight, we move forward with our investigation of faces to the right side of the image, and remove the faces discovered.

As demonstrated above, our observation reveals the existence of 7 people.Considering all factors, the cumulative count of individuals in the image is 27.

The final answer is: 27


How many people can you identify in the image? option:[8, 18, 28, 38]

To start off, let's eliminate the faces on the left side of the image.

As demonstrated above, our observation reveals the existence of 10 people.Building upon this, we proceed to the right in the image, searching for faces and removing them then.

The image above reveals our identification of 10 individuals.With this information in mind, we persist in our quest for faces to the right in the image, eliminating the faces discovered then.

The visual reference above confirms our identification of 10 individuals.Based on this premise, we advance our examination of faces to the right in the image, eliminating the faces identified.

The visual reference above confirms our identification of 8 individuals.Taking everything into consideration, the image contains a collective count of 38 people.

The final answer is: 38


How many people are visible in the image? option:[26, 36, 46, 56]

The visual reference above confirms our identification of 10 individuals.Taking this into consideration, we extend our examination of faces to the right in the image, deleting them then.

As demonstrated above, our observation reveals the existence of 10 people.Expanding on this, we proceed to the right in the image, seeking faces and removing them.

As demonstrated above, our observation reveals the existence of 10 people.In light of this, we continue the search for faces to the right in the image, then deleting the faces located.

The above illustration illustrates our discovery of a total of 6 people.To conclude, the total number of people in the image amounts to 46.

The final answer is: 46


How many people are there in the image? option:[11, 21, 31, 41]

Our first step involves deleting the faces visible on the left side of the image.

In the depicted image, our findings indicate the presence of 10 individuals.Leveraging this insight, we move forward with our investigation of faces to the right side of the image, and remove the faces discovered.

As demonstrated above, our observation reveals the existence of 10 people.With this information in mind, we persist in our quest for faces to the right in the image, eliminating the faces discovered then.

In the image above, we've successfully located 10 people.Taking this into consideration, we extend our examination of faces to the right in the image, deleting them then.

As demonstrated above, our observation reveals the existence of 1 people.In summation, the total headcount in the image adds up to 31 people.

The final answer is: 31


How many people do you see in the image? option:[27, 37, 47, 57]

Our initial action involves removing the faces situated on the left side of the image.

As demonstrated above, our observation reveals the existence of 10 people.Taking this into consideration, we extend our examination of faces to the right in the image, deleting them then.

Based on the visual representation above, we've pinpointed a count of 7 people.Summing up, there are a combined 27 persons depicted in the image.

The final answer is: 27


Can you ascertain the quantity of people captured in the image? option:[5, 15, 25, 35]

Our initial action involves removing the faces on the left side of the image.

The image above reveals our identification of 10 individuals.Given this context, we move on to the right in the image, searching for faces and eliminating them.

As demonstrated above, our observation reveals the existence of 10 people.Given this context, we move on to the right in the image, searching for faces and eliminating them.

The visual reference above confirms our identification of 5 individuals.Putting it all together, the comprehensive count of individuals in the image is 25.

The final answer is: 25


Can you ascertain the quantity of people captured in the image? option:[2, 12, 22, 32]

As shown in the above image, we have identified a total of 10 people.Given this context, we move on to the right in the image, searching for faces and eliminating them.

In the depicted image, our findings indicate the presence of 10 individuals.Taking this into consideration, we extend our examination of faces to the right in the image, deleting them then.

According to the illustration above, we've discovered a group of 2 individuals.Summing up, there are a combined 22 persons depicted in the image.

The final answer is: 22


Would you be able to figure out how many people are in the image? option:[33, 43, 53, 63]

We initiate the process by deleting the faces on the left of the image.

According to the illustration above, we've discovered a group of 10 individuals.Building upon this, we proceed to the right in the image, searching for faces and removing them then.

The visual reference above confirms our identification of 10 individuals.Taking this into consideration, we extend our examination of faces to the right in the image, deleting them then.

The image above reveals our identification of 10 individuals.In light of this, we continue the search for faces to the right in the image, then deleting the faces located.

The image above reveals our identification of 10 individuals.With this information in mind, we persist in our quest for faces to the right in the image, eliminating the faces discovered then.

As shown in the above image, we have identified a total of 3 people.In summary, there are a total of 43 individuals in the image.

The final answer is: 43


Can you give a count of the people present in the image? option:[47, 57, 67, 77]

We'll commence by deleting the faces on the left of the image.

In the image above, we've successfully located 10 people.Taking this into consideration, we extend our examination of faces to the right in the image, deleting them then.

The image above reveals our identification of 10 individuals.Given this, we extend our search for faces to the right in the image, and delete the faces that are found.

The visual reference above confirms our identification of 10 individuals.With this information in mind, we persist in our quest for faces to the right in the image, eliminating the faces discovered then.

As shown in the above image, we have identified a total of 10 people.Taking this into consideration, we extend our examination of faces to the right in the image, deleting them then.

In the image above, we've successfully located 7 people.To conclude, the total number of people in the image amounts to 57.

The final answer is: 57


How many people do you see in the image? option:[16, 26, 36, 46]

Our initial action involves removing the faces situated on the left side of the image.

As demonstrated above, our observation reveals the existence of 10 people.Building upon this, we proceed to the right in the image, searching for faces and removing them then.

In the image above, we've successfully located 10 people.Based on this premise, we advance our examination of faces to the right in the image, eliminating the faces identified.

Our analysis of the above image indicates the presence of 6 individuals.Putting it all together, the comprehensive count of individuals in the image is 26.

The final answer is: 26


Can you count the number of people present in the image? option:[27, 37, 47, 57]

Our analysis of the above image indicates the presence of 10 individuals.In light of this, we continue the search for faces to the right in the image, then deleting the faces located.

Our analysis of the above image indicates the presence of 7 individuals.To summarize, the overall number of individuals present in the image is 27.

The final answer is: 27


Could you provide a count of the people in the given image? option:[13, 23, 33, 43]

We initiate the process by deleting the faces on the left of the image.

According to the illustration above, we've discovered a group of 10 individuals.Leveraging this insight, we move forward with our investigation of faces to the right side of the image, and remove the faces discovered.

Our analysis of the above image indicates the presence of 3 individuals.Considering all factors, the cumulative count of individuals in the image is 23.

The final answer is: 23


How many people do you see in the image? option:[20, 30, 40, 50]

Our analysis of the above image indicates the presence of 10 individuals.Taking this into consideration, we extend our examination of faces to the right in the image, deleting them then.

The above illustration illustrates our discovery of a total of 10 people.Given this, we extend our search for faces to the right in the image, and delete the faces that are found.

According to the illustration above, we've discovered a group of 10 individuals.Given this, we extend our search for faces to the right in the image, and delete the faces that are found.

As shown in the above image, we have identified a total of 0 people.Summing up, there are a combined 30 persons depicted in the image.

The final answer is: 30


Could you provide a count of the people in the given image? option:[12, 22, 32, 42]

As a starting point, we'll initiate the process by eliminating the faces positioned on the left side of the image.

Our analysis of the above image indicates the presence of 10 individuals.Considering this development, we extend our search for faces to the right side of the image, and delete the faces found.

In the depicted image, our findings indicate the presence of 10 individuals.With this information in mind, we persist in our quest for faces to the right in the image, eliminating the faces discovered then.

As shown in the above image, we have identified a total of 10 people.Taking this into consideration, we extend our examination of faces to the right in the image, deleting them then.

The visual reference above confirms our identification of 2 individuals.In summation, the total headcount in the image adds up to 32 people.

The final answer is: 32


How many people can you identify in the image? option:[20, 30, 40, 50]

Our initial action involves removing the faces on the left side of the image.

In the image above, we've successfully located 10 people.Considering this development, we extend our search for faces to the right side of the image, and delete the faces found.

According to the illustration above, we've discovered a group of 10 individuals.Given this context, we move on to the right in the image, searching for faces and eliminating them.

The above illustration illustrates our discovery of a total of 10 people.Based on this premise, we advance our examination of faces to the right in the image, eliminating the faces identified.

In the depicted image, our findings indicate the presence of 10 individuals.Considering this development, we extend our search for faces to the right side of the image, and delete the faces found.

As demonstrated above, our observation reveals the existence of 0 people.Considering all factors, the cumulative count of individuals in the image is 40.

The final answer is: 40


How many people do you see in the image? option:[1, 11, 21, 31]

As demonstrated above, our observation reveals the existence of 10 people.Taking this into consideration, we extend our examination of faces to the right in the image, deleting them then.

The visual reference above confirms our identification of 1 individuals.In conclusion, there are 31 people in total captured in the image.

The final answer is: 31


How many people are visible in the image? option:[30, 40, 50, 60]

Our analysis of the above image indicates the presence of 10 individuals.In light of this, we continue the search for faces to the right in the image, then deleting the faces located.

According to the illustration above, we've discovered a group of 10 individuals.Considering this development, we extend our search for faces to the right side of the image, and delete the faces found.

In the image above, we've successfully located 10 people.Expanding on this, we proceed to the right in the image, seeking faces and removing them.

The image above reveals our identification of 0 individuals.In summary, there are a total of 40 individuals in the image.

The final answer is: 40


Would you be able to figure out how many people are in the image? option:[16, 26, 36, 46]

As a starting point, we'll initiate the process by eliminating the faces positioned on the left side of the image.

In the depicted image, our findings indicate the presence of 10 individuals.Given this, we extend our search for faces to the right in the image, and delete the faces that are found.

Based on the visual representation above, we've pinpointed a count of 10 people.Building upon this, we proceed to the right in the image, searching for faces and removing them then.

As demonstrated above, our observation reveals the existence of 10 people.Leveraging this insight, we move forward with our investigation of faces to the right side of the image, and remove the faces discovered.

Our analysis of the above image indicates the presence of 10 individuals.Expanding on this, we proceed to the right in the image, seeking faces and removing them.

According to the illustration above, we've discovered a group of 6 individuals.In summary, there are a total of 46 individuals in the image.

The final answer is: 46


Could you provide a count of the people in the given image? option:[21, 31, 41, 51]

We'll commence by deleting the faces on the left of the image.

In the image above, we've successfully located 10 people.Given this, we extend our search for faces to the right in the image, and delete the faces that are found.

The image above reveals our identification of 1 individuals.In conclusion, there are 21 people in total captured in the image.

The final answer is: 21


Can you ascertain the quantity of people captured in the image? option:[35, 45, 55, 65]

Our initial action involves removing the faces situated on the left side of the image.

As shown in the above image, we have identified a total of 10 people.Leveraging this insight, we move forward with our investigation of faces to the right side of the image, and remove the faces discovered.

The above illustration illustrates our discovery of a total of 10 people.Expanding on this, we proceed to the right in the image, seeking faces and removing them.

The above illustration illustrates our discovery of a total of 10 people.Leveraging this insight, we move forward with our investigation of faces to the right side of the image, and remove the faces discovered.

According to the illustration above, we've discovered a group of 10 individuals.Given this, we extend our search for faces to the right in the image, and delete the faces that are found.

Based on the visual representation above, we've pinpointed a count of 5 people.In summary, there are a total of 45 individuals in the image.

The final answer is: 45


Can you count the number of people appearing in the image? option:[0, 10, 20, 30]

To begin with, our first step will be to erase the faces located on the left side of the image.

Based on the visual representation above, we've pinpointed a count of 10 people.Based on this premise, we advance our examination of faces to the right in the image, eliminating the faces identified.

The above illustration illustrates our discovery of a total of 10 people.With this information in mind, we persist in our quest for faces to the right in the image, eliminating the faces discovered then.

The visual reference above confirms our identification of 10 individuals.Taking this into consideration, we extend our examination of faces to the right in the image, deleting them then.

Based on the visual representation above, we've pinpointed a count of 0 people.In conclusion, there are 30 people in total captured in the image.

The final answer is: 30


Initially, we will remove the faces on the left side of the image.

The above illustration illustrates our discovery of a total of 10 people.Given this, we extend our search for faces to the right in the image, and delete the faces that are found.

In the depicted image, our findings indicate the presence of 10 individuals.Based on this premise, we advance our examination of faces to the right in the image, eliminating the faces identified.

Based on the visual representation above, we've pinpointed a count of 2 people.Putting it all together, the comprehensive count of individuals in the image is 22.

The final answer is: 22


We initiate the process by deleting the faces on the left of the image.

Our analysis of the above image indicates the presence of 10 individuals.Building upon this, we proceed to the right in the image, searching for faces and removing them then.

The above illustration illustrates our discovery of a total of 10 people.In light of this, we continue the search for faces to the right in the image, then deleting the faces located.

The image above reveals our identification of 10 individuals.Leveraging this insight, we move forward with our investigation of faces to the right side of the image, and remove the faces discovered.

As demonstrated above, our observation reveals the existence of 10 people.With this information in mind, we persist in our quest for faces to the right in the image, eliminating the faces discovered then.

The above illustration illustrates our discovery of a total of 10 people.Taking this into consideration, we extend our examination of faces to the right in the image, deleting them then.

Based on the visual representation above, we've pinpointed a count of 7 people.Taking everything into consideration, the image contains a collective count of 57 people.

The final answer is: 57


Can you count the number of people present in the image? option:[36, 46, 56, 66]

Initially, our focus will be on removing the faces located on the left side of the image.

Our analysis of the above image indicates the presence of 10 individuals.Leveraging this insight, we move forward with our investigation of faces to the right side of the image, and remove the faces discovered.

Our analysis of the above image indicates the presence of 10 individuals.Expanding on this, we proceed to the right in the image, seeking faces and removing them.

The visual reference above confirms our identification of 10 individuals.Considering this development, we extend our search for faces to the right side of the image, and delete the faces found.

The visual reference above confirms our identification of 10 individuals.Expanding on this, we proceed to the right in the image, seeking faces and removing them.

Our analysis of the above image indicates the presence of 10 individuals.Considering this development, we extend our search for faces to the right side of the image, and delete the faces found.

As shown in the above image, we have identified a total of 6 people.In summation, the total headcount in the image adds up to 56 people.

The final answer is: 56


How many people do you see in the image? option:[22, 32, 42, 52]

Our initial action involves removing the faces situated on the left side of the image.

As demonstrated above, our observation reveals the existence of 10 people.Expanding on this, we proceed to the right in the image, seeking faces and removing them.

The image above reveals our identification of 10 individuals.Taking this into consideration, we extend our examination of faces to the right in the image, deleting them then.

The image above reveals our identification of 10 individuals.Based on this premise, we advance our examination of faces to the right in the image, eliminating the faces identified.

Based on the visual representation above, we've pinpointed a count of 10 people.Given this context, we move on to the right in the image, searching for faces and eliminating them.

The visual reference above confirms our identification of 2 individuals.In overview, the image showcases a grand total of 52 individuals.

The final answer is: 52


How many people are there in the image? option:[16, 26, 36, 46]

To begin with, our first step will be to erase the faces located on the left side of the image.

In the image above, we've successfully located 10 people.Given this, we extend our search for faces to the right in the image, and delete the faces that are found.

Based on the visual representation above, we've pinpointed a count of 10 people.In light of this, we continue the search for faces to the right in the image, then deleting the faces located.

As demonstrated above, our observation reveals the existence of 10 people.Taking this into consideration, we extend our examination of faces to the right in the image, deleting them then.

In the image above, we've successfully located 10 people.Considering this development, we extend our search for faces to the right side of the image, and delete the faces found.

As demonstrated above, our observation reveals the existence of 6 people.In summation, the total headcount in the image adds up to 46 people.

The final answer is: 46


How many people are visible in the image? option:[40, 50, 60, 70]

Our first step involves deleting the faces visible on the left side of the image.

The visual reference above confirms our identification of 10 individuals.Considering this development, we extend our search for faces to the right side of the image, and delete the faces found.

According to the illustration above, we've discovered a group of 10 individuals.Given this, we extend our search for faces to the right in the image, and delete the faces that are found.

As demonstrated above, our observation reveals the existence of 10 people.In light of this, we continue the search for faces to the right in the image, then deleting the faces located.

In the image above, we've successfully located 10 people.Given this context, we move on to the right in the image, searching for faces and eliminating them.

In the depicted image, our findings indicate the presence of 10 individuals.In light of this, we continue the search for faces to the right in the image, then deleting the faces located.

Based on the visual representation above, we've pinpointed a count of 0 people.Considering all factors, the cumulative count of individuals in the image is 50.

The final answer is: 50


Can you count the number of people appearing in the image? option:[22, 32, 42, 52]

To start off, let's eliminate the faces on the left side of the image.

As demonstrated above, our observation reveals the existence of 10 people.Building upon this, we proceed to the right in the image, searching for faces and removing them then.

As demonstrated above, our observation reveals the existence of 10 people.Considering this development, we extend our search for faces to the right side of the image, and delete the faces found.

The image above reveals our identification of 10 individuals.Taking this into consideration, we extend our examination of faces to the right in the image, deleting them then.

According to the illustration above, we've discovered a group of 2 individuals.To conclude, the total number of people in the image amounts to 32.

The final answer is: 32


How many people are there in the image? option:[36, 46, 56, 66]

We initiate the process by deleting the faces on the left of the image.

In the depicted image, our findings indicate the presence of 10 individuals.Expanding on this, we proceed to the right in the image, seeking faces and removing them.

In the depicted image, our findings indicate the presence of 10 individuals.With this information in mind, we persist in our quest for faces to the right in the image, eliminating the faces discovered then.

In the depicted image, our findings indicate the presence of 10 individuals.Considering this development, we extend our search for faces to the right side of the image, and delete the faces found.

According to the illustration above, we've discovered a group of 10 individuals.With this information in mind, we persist in our quest for faces to the right in the image, eliminating the faces discovered then.

As demonstrated above, our observation reveals the existence of 6 people.Putting it all together, the comprehensive count of individuals in the image is 46.

The final answer is: 46


Can you count the number of people present in the image? option:[10, 20, 30, 40]

As demonstrated above, our observation reveals the existence of 10 people.Based on this premise, we advance our examination of faces to the right in the image, eliminating the faces identified.

In the depicted image, our findings indicate the presence of 10 individuals.Given this, we extend our search for faces to the right in the image, and delete the faces that are found.

Our analysis of the above image indicates the presence of 10 individuals.Given this context, we move on to the right in the image, searching for faces and eliminating them.

In the image above, we've successfully located 0 people.Summing up, there are a combined 40 persons depicted in the image.

The final answer is: 40


How many people do you see in the image? option:[33, 3, 13, 23]

Our first step involves deleting the faces visible on the left side of the image.

According to the illustration above, we've discovered a group of 10 individuals.Based on this premise, we advance our examination of faces to the right in the image, eliminating the faces identified.

The visual reference above confirms our identification of 10 individuals.Considering this development, we extend our search for faces to the right side of the image, and delete the faces found.

The visual reference above confirms our identification of 3 individuals.Summing up, there are a combined 23 persons depicted in the image.

The final answer is: 23


Can you count the number of people present in the image? option:[15, 25, 35, 45]

As a starting point, we'll initiate the process by eliminating the faces positioned on the left side of the image.

The image above reveals our identification of 10 individuals.Based on this premise, we advance our examination of faces to the right in the image, eliminating the faces identified.

Our analysis of the above image indicates the presence of 10 individuals.Leveraging this insight, we move forward with our investigation of faces to the right side of the image, and remove the faces discovered.

As shown in the above image, we have identified a total of 5 people.Putting it all together, the comprehensive count of individuals in the image is 25.

The final answer is: 25


Can you ascertain the quantity of people captured in the image? option:[43, 53, 63, 73]

Our initial action involves removing the faces on the left side of the image.

The above illustration illustrates our discovery of a total of 10 people.Given this, we extend our search for faces to the right in the image, and delete the faces that are found.

In the depicted image, our findings indicate the presence of 10 individuals.Leveraging this insight, we move forward with our investigation of faces to the right side of the image, and remove the faces discovered.

As demonstrated above, our observation reveals the existence of 10 people.Building upon this, we proceed to the right in the image, searching for faces and removing them then.

As shown in the above image, we have identified a total of 10 people.Building upon this, we proceed to the right in the image, searching for faces and removing them then.

In the image above, we've successfully located 10 people.Expanding on this, we proceed to the right in the image, seeking faces and removing them.

In the depicted image, our findings indicate the presence of 3 individuals.In overview, the image showcases a grand total of 53 individuals.

The final answer is: 53


Can you ascertain the quantity of people captured in the image? option:[0, 10, 20, 30]

Based on the visual representation above, we've pinpointed a count of 10 people.Considering this development, we extend our search for faces to the right side of the image, and delete the faces found.

In the depicted image, our findings indicate the presence of 10 individuals.Considering this development, we extend our search for faces to the right side of the image, and delete the faces found.

According to the illustration above, we've discovered a group of 0 individuals.Putting it all together, the comprehensive count of individuals in the image is 30.

The final answer is: 30


Can you count the number of people present in the image? option:[1, 11, 21, 31]

To start off, let's eliminate the faces on the left side of the image.

In the image above, we've successfully located 10 people.Given this, we extend our search for faces to the right in the image, and delete the faces that are found.

As shown in the above image, we have identified a total of 1 people.Summing up, there are a combined 21 persons depicted in the image.

The final answer is: 21


How many people can you identify in the image? option:[29, 39, 49, 59]

Our first step involves deleting the faces visible on the left side of the image.

Our analysis of the above image indicates the presence of 10 individuals.Building upon this, we proceed to the right in the image, searching for faces and removing them then.

According to the illustration above, we've discovered a group of 10 individuals.Based on this premise, we advance our examination of faces to the right in the image, eliminating the faces identified.

Based on the visual representation above, we've pinpointed a count of 10 people.With this information in mind, we persist in our quest for faces to the right in the image, eliminating the faces discovered then.

The above illustration illustrates our discovery of a total of 9 people.Summing up, there are a combined 39 persons depicted in the image.

The final answer is: 39


Can you count the number of people appearing in the image? option:[59, 69, 79, 89]

Our initial action involves removing the faces on the left side of the image.

As shown in the above image, we have identified a total of 10 people.With this information in mind, we persist in our quest for faces to the right in the image, eliminating the faces discovered then.

According to the illustration above, we've discovered a group of 10 individuals.Expanding on this, we proceed to the right in the image, seeking faces and removing them.

In the image above, we've successfully located 10 people.Building upon this, we proceed to the right in the image, searching for faces and removing them then.

The visual reference above confirms our identification of 10 individuals.Based on this premise, we advance our examination of faces to the right in the image, eliminating the faces identified.

As demonstrated above, our observation reveals the existence of 10 people.Based on this premise, we advance our examination of faces to the right in the image, eliminating the faces identified.

As shown in the above image, we have identified a total of 9 people.Considering all factors, the cumulative count of individuals in the image is 59.

The final answer is: 59


Can you give a count of the people present in the image? option:[27, 37, 47, 57]

Initially, our focus will be on removing the faces located on the left side of the image.

The visual reference above confirms our identification of 10 individuals.Expanding on this, we proceed to the right in the image, seeking faces and removing them.

The visual reference above confirms our identification of 10 individuals.Given this context, we move on to the right in the image, searching for faces and eliminating them.

In the image above, we've successfully located 10 people.Expanding on this, we proceed to the right in the image, seeking faces and removing them.

According to the illustration above, we've discovered a group of 7 individuals.In summation, the total headcount in the image adds up to 37 people.

The final answer is: 37


How many people are visible in the image? option:[38, 8, 18, 28]

Initially, we will remove the faces on the left side of the image.

The above illustration illustrates our discovery of a total of 10 people.Building upon this, we proceed to the right in the image, searching for faces and removing them then.

Our analysis of the above image indicates the presence of 10 individuals.In light of this, we continue the search for faces to the right in the image, then deleting the faces located.

As demonstrated above, our observation reveals the existence of 8 people.To conclude, the total number of people in the image amounts to 28.

The final answer is: 28


Can you count the number of people appearing in the image? option:[23, 33, 43, 53]

Our first step involves deleting the faces visible on the left side of the image.

According to the illustration above, we've discovered a group of 10 individuals.Considering this development, we extend our search for faces to the right side of the image, and delete the faces found.

Based on the visual representation above, we've pinpointed a count of 10 people.Leveraging this insight, we move forward with our investigation of faces to the right side of the image, and remove the faces discovered.

The above illustration illustrates our discovery of a total of 10 people.With this information in mind, we persist in our quest for faces to the right in the image, eliminating the faces discovered then.

The above illustration illustrates our discovery of a total of 3 people.In conclusion, there are 33 people in total captured in the image.

The final answer is: 33


Can you give a count of the people present in the image? option:[20, 30, 40, 50]

According to the illustration above, we've discovered a group of 10 individuals.Based on this premise, we advance our examination of faces to the right in the image, eliminating the faces identified.

The image above reveals our identification of 10 individuals.Given this context, we move on to the right in the image, searching for faces and eliminating them.

Our analysis of the above image indicates the presence of 10 individuals.Given this context, we move on to the right in the image, searching for faces and eliminating them.

Based on the visual representation above, we've pinpointed a count of 10 people.Given this, we extend our search for faces to the right in the image, and delete the faces that are found.

As demonstrated above, our observation reveals the existence of 0 people.In summary, there are a total of 50 individuals in the image.

The final answer is: 50


As demonstrated above, our observation reveals the existence of 10 people.Expanding on this, we proceed to the right in the image, seeking faces and removing them.

Our analysis of the above image indicates the presence of 10 individuals.Building upon this, we proceed to the right in the image, searching for faces and removing them then.

As demonstrated above, our observation reveals the existence of 10 people.Given this context, we move on to the right in the image, searching for faces and eliminating them.

The image above reveals our identification of 10 individuals.With this information in mind, we persist in our quest for faces to the right in the image, eliminating the faces discovered then.

Based on the visual representation above, we've pinpointed a count of 0 people.Taking everything into consideration, the image contains a collective count of 40 people.

The final answer is: 40


Could you provide a count of the people in the given image? option:[33, 43, 53, 63]

As a starting point, we'll initiate the process by eliminating the faces positioned on the left side of the image.

The image above reveals our identification of 10 individuals.Based on this premise, we advance our examination of faces to the right in the image, eliminating the faces identified.

As shown in the above image, we have identified a total of 10 people.Expanding on this, we proceed to the right in the image, seeking faces and removing them.

The visual reference above confirms our identification of 10 individuals.Given this, we extend our search for faces to the right in the image, and delete the faces that are found.

According to the illustration above, we've discovered a group of 3 individuals.Taking everything into consideration, the image contains a collective count of 33 people.

The final answer is: 33


Can you count the number of people appearing in the image? option:[36, 6, 16, 26]

According to the illustration above, we've discovered a group of 10 individuals.Taking this into consideration, we extend our examination of faces to the right in the image, deleting them then.

As shown in the above image, we have identified a total of 6 people.Taking everything into consideration, the image contains a collective count of 26 people.

The final answer is: 26


Can you count the number of people appearing in the image? option:[34, 44, 54, 64]

To begin with, our first step will be to erase the faces located on the left side of the image.

Our analysis of the above image indicates the presence of 10 individuals.Taking this into consideration, we extend our examination of faces to the right in the image, deleting them then.

Our analysis of the above image indicates the presence of 10 individuals.Leveraging this insight, we move forward with our investigation of faces to the right side of the image, and remove the faces discovered.

The above illustration illustrates our discovery of a total of 10 people.Given this context, we move on to the right in the image, searching for faces and eliminating them.

In the image above, we've successfully located 4 people.In summary, there are a total of 44 individuals in the image.

The final answer is: 44


Can you give a count of the people present in the image? option:[18, 28, 38, 48]

Our first step involves deleting the faces visible on the left side of the image.

According to the illustration above, we've discovered a group of 10 individuals.Leveraging this insight, we move forward with our investigation of faces to the right side of the image, and remove the faces discovered.

According to the illustration above, we've discovered a group of 10 individuals.With this information in mind, we persist in our quest for faces to the right in the image, eliminating the faces discovered then.

In the image above, we've successfully located 10 people.With this information in mind, we persist in our quest for faces to the right in the image, eliminating the faces discovered then.

Our analysis of the above image indicates the presence of 8 individuals.To conclude, the total number of people in the image amounts to 48.

The final answer is: 48