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pgs 7/10/07 8:56 AM Page 518 518 Ratio, Proportion, and Similarity x s x When a diagonal of a square is drawn, the square is separated into two isosceles right triangles. We can express the length of a leg of the isosceles right triangle in terms of the length of the hypotenuse or the length of the hypotenuse in terms ... |
the hypotenuse,, is 6 centimeters and the length of one leg is 3 cenAB timeters. Find the length of the other leg. B a2 1 b2 5 c2 a2 1 (3)2 5 (6)2 a2 1 9 5 36 a2 5 27 a 5 27 5 " 6 cm 3 Answer C 3 cm A 9? 3 5 3 " " Note: The measure of one leg is one-half the measure of the hypotenuse and the length of the other leg is... |
rhombus is 10 centimeters and the length of one diagonal is 120 millimeters. Find the length of the other diagonal. 12. The length of each side of a rhombus is 13 feet. If the length of the shorter diagonal is 10 feet, find the length of the longer diagonal. 13. Find the length of the diagonal of a rectangle whose sid... |
pgs 7/10/07 8:56 AM Page 521 The Distance Formula 521 23. A young tree is braced by wires that are 9 feet long and fastened at a point on the trunk of the tree 5 feet from the ground. Find to the nearest tenth of a foot how far from the foot of the tree the wires should be fastened to the ground in order to be sure tha... |
O A(x2, y2) c b a B(x1, y1) C(x2, y1) x c2 a2 b2 x1 c2 x2 c (x2 2 x1)2 1 (y2 2 y1)2 2 y2 y1 2 " This result is called the distance formula. If the endpoints of a line segment in the coordinate plane are B(x1, y1) and A(x2, y2), then: AB " (x2 2 x1)2 1 (y2 2 y1)2 EXAMPLE 1 The coordinates of the vertices of quadrilater... |
and ABCD is not a square. METHOD 2 slope of AB 5 24 2 (23) 6 2 (21) slope of BC 5 3 2 (24) 5 2 6 5 21 7 5 21 7 5 7 21 5 27 AB The slope of Therefore, AB rhombus is not a square. is not equal to the negative reciprocal of the slope of. BC, B is not a right angle and the is not perpendicular to BC EXAMPLE 2 Prove that t... |
(2a, 0) x 22a 1 0 2, 0 1 2b 2 A The length of B CM 5 (2a, b). is 2a 1 0 2, 0 1 2b 2 A The length of B AN is 5 (a, b). (2a 2 2a)2 1 (b 2 0)2 (22a 2 a)2 1 (0 2 b)2 " 5 (23a)2 1 b2 " " 5 (23a)2 1 (2b)2 " 5 9a2 1 b2 " CM AN; therefore, CM > AN. 5 9a2 1 b2 " Exercises Writing About Mathematics 1. Can the distance formula be... |
. The vertices of a triangle are P(1, 1), Q(7, 1), and R(3, 3). a. Show that PQR is an isosceles triangle. b. Show that PQR is a right triangle using the Pythagorean Theorem. c. Show that the midpoint of the hypotenuse is equidistant from the vertices. 15. The vertices of a triangle are L(1, 1), M(7, 3), and N(2, 2). a... |
vertices of quadrilateral ABCD, the image of ABCD under the transformation D3. c. Show that ABCD is a parallelogram using the distance formula. d. Use part c to show that the images of the parallel segments of ABCD are also parallel under the dilation. 21. The vertices of quadrilateral ABCD are A(2, 2), B(2, 0), C(3, ... |
of two numbers, a and b, where b is not zero, is the number. b • A proportion is an equation that states that two ratios are equal. Chapter Summary 527 • In the proportion, the first and fourth terms, a and d, are the extremes of the proportion, and the second and third terms, b and c, are the means. b 5 c a d • If th... |
the product of the means is equal to the product of the extremes. In a proportion, the means may be interchanged. 12.1a 12.1b In a proportion, the extremes may be interchanged. 12.1c If the products of two pairs of factors are equal, the factors of one pair can be the means and the factors of the other the extremes of... |
hypotenuse. 12.16b The length of the altitude to the hypotenuse of a right triangle is the mean proportional between the lengths of the projections of the legs on the hypotenuse. 12.17 A triangle is a right triangle if and only if the square of the length of the longest side is equal to the sum of the squares of the l... |
. Are the triangles similar? Justify your answer. 14365C12.pgs 7/10/07 8:56 AM Page 530 530 Ratio, Proportion, and Similarity 3. A line parallel to side AB of ABC intersects AC at E and BC at F. If EC 12, AC 20, and AB 15, find EF. 4. A line intersects side AC FC 5, BC 15, prove that EFC ABC. of ABC at E and BC at F. I... |
13. Find the length of a side of a rhombus if the measures of the diagonals of the rhombus are 30 inches and 40 inches. 14365C12.pgs 7/10/07 8:56 AM Page 531 Review Exercises 531 14. The length of a side of a rhombus is 26.0 centimeters and the length of one diagonal is 28.0 centimeters. Find to the nearest tenth the ... |
. 5 " B 14365C12.pgs 7/10/07 8:56 AM Page 532 532 Ratio, Proportion, and Similarity Follow the steps to construct a golden rectangle. You may use compass and straightedge or geometry software. STEP 1. Draw square ABCD. STEP 2. Construct E, the midpoint of AB. STEP 3. Draw the ray h AB. STEP 4. With E as the center and ... |
(1) three lines each perpendicular to the other two (2) two parallel lines (4) a line and a point not on it (3) two intersecting lines 6. Which of the following cannot be the measures of the sides of a triangle? (1) 5, 7, 8 (2) 3, 8, 9 (3) 5, 7, 7 (4) 2, 6, 8 7. Under a rotation of 90° about the origin, the image of t... |
a right circular cone is one-half the slant height of the cone. The radius of the base is 2.50 feet. a. Find the lateral area of the cone to the nearest tenth of a square foot. b. Find the volume of the cone to the nearest tenth of a cubic foot. 14. The coordinates of the vertices of ABC are A(1, 0), B(4, 0), and C(2,... |
to one another? Archimedes (287–212 B.C.) proposed that the area of a circle was equal to the area of a right triangle whose legs have lengths equal to the radius, r, and the circumference, C, of a circle. Thus A rC. He used indirect proof and the areas of inscribed and circumscribed polygons to prove his conjecture a... |
from its center, OA OB OC. Thus, because equal line segments are congruent. We can state what we have just proved as a theorem. OA > OB > OC, and, OA OB OC O A C Theorem 13.1 All radii of the same circle are congruent. A circle separates a plane into three sets of points. If we let the length of the radius of circle O... |
X each of which is called a semicircle. In the diagram above, name two different semicircles. and. ADBX ADCX Degree Measure of an Arc An arc of a circle is called an intercepted arc, or an arc intercepted by an angle, if each endpoint of the arc is on a different ray of the angle and the other points of the arc are in ... |
> (Or CsDsX CDX > CrDrX is not congruent to. However, if circle CDX CsDsX even if D 60 C O C O 60 D 60 C D O Postulate 13.1 Arc Addition Postulate BCX and mABCX BCX ABX If point and no other points in common, then ABX m. m are two arcs of the same circle having a common endand ABCXBCX ABX The arc that is the sum of tw... |
and. The converse of this theorem can be proved by using the same definitions and postulates. Theorem 13.2b In a circle or in congruent circles, central angles are congruent if their intercepted arcs are congruent. Theorems 13.2a and 13.2b can be written as a biconditional. Theorem 13.2 In a circle or in congruent cir... |
Find each mABX 14. 16. mDOA 18. mBOD mDABX 22. mADCX 20. C 89° O 42° D B A 14541C13.pgs 1/25/08 3:48 PM Page 542 542 Geometry of the Circle 23. In 23–32, P, Q, S, and R are points on circle O, mPOQ 100, mQOS 110, and mSOR 35. Find each measure. mPQX mSRX mRPX 27. 29. mQOR mSRPX mQSX 24. 26. mROP mPQSX mQSRX mRPQX 110°... |
radius of circle O is r, and the AOC AOC AOC length of the diameter is d, then d 5 AOC 5 OA 1 OC 5 r 1 r 5 2r d 2r That is: ABX, and major The endpoints of a chord are points on a circle and, therefore, determine two arcs of a circle, a minor arc and, central AOB, a major arc. In the diagram, chord are all determined ... |
a biconditional. Theorem 13.3 In a circle or in congruent circles, two chords are congruent if and only if their central angles are congruent. Since central angles and their intercepted arcs have equal degree measures, we can also prove the following theorems. Theorem 13.4a In a circle or in congruent circles, congrue... |
Therefore, since mABX mCDX their arcs are congruent. Therefore, 35 and ABX 35, CDX AB > CD and AB = CD.. In a circle, chords are congruent if 14365C13.pgs 7/12/07 3:56 PM Page 546 546 Geometry of the Circle Chords Equidistant from the Center of a Circle We defined the distance from a point to a line as the length of t... |
arcs. An apothem of a circle is a perpendicular line segment from the center of a circle to the midpoint of a chord. The term apothem also refers to the length of the segment. In the diagram, E is the midpoint of chord in circle O, AB'CD, or OE, is the apothem., and AB OE A C E O D Theorem 13.6 The perpendicular bisec... |
CD are radii of the same. Therefore, OBE ODF by HL, and and OD. A line through the center of a circle that is perpendicular to a OB OD. Since and OB OB > OD Proof Draw circle, EB > FD chord bisects the chord. Thus, gruent segments are congruent, AE > EB and AB > CD. CF > FD. Since doubles of con- A C E O F B D We can ... |
2 2 FD2 OE2. OF2 OE. OF Therefore, the shorter chord is farther from the center of the circle. We have just proved the following theorem: Theorem 13.8 In a circle, if the lengths of two chords are unequal, then the shorter chord is farther from the center. EXAMPLE 2 In circle O, mABX 90 and OA 6. a. Prove that AOB is a... |
ectors of the sides of ABC. Every point on the perpendicular bisectors of a line segment is equidistant from the endpoints of the line segment. Therefore, PA PB PC and A, B, and C are points on a circle with center at P, that is, any triangle can be inscribed in a circle. g, PL g PM, and C P L M B N A 14365C13.pgs 7/12... |
is a diameter of circle O, AB is a chord of the circle, and OD'AB at C. 14. If AB 8 and OC 3, find OB. 16. If OC 20 and OB 25, find AB. 15. If AB 48 and OC 7, find OB. 17. If OC 12 and OB 18, find AB. 14365C13.pgs 7/12/07 3:56 PM Page 552 552 Geometry of the Circle 18. If AB 18 and OB 15, find OC. 19. If AB 20 and OB ... |
of its arc to find the relationship between ABC and the measure of its arc,.ACX 14365C13.pgs 7/12/07 3:56 PM Page 553 CASE 1 One of the sides of the inscribed angle contains a diameter of the circle. Inscribed Angles and Their Measures 553 B x x O 2x A 2x C OA Consider first an inscribed angle, ABC, with a diameter of... |
pgs 7/12/07 3:56 PM Page 554 554 Geometry of the Circle Corollary 13.9a An angle inscribed in a semicircle is a right angle. Proof: In the diagram, ABC is inscribed in semicircle semicircle whose degree measure is 180°. Therefore, is a diameter of circle O, and is a. Also ADCX ABCX AOC mABC 2 mADCX 5 1 5 1 2(180) 5 90 ... |
m/1 5 1 2m ABX Exercises Writing About Mathematics 1. Explain how you could use Corollary 13.9a to construct a right triangle with two given line segments as the hypotenuse and one leg. 14365C13.pgs 7/12/07 3:56 PM Page 556 556 Geometry of the Circle 2. In circle O, ABC is an inscribed angle and mACX 50. In circle O, ... |
f. mC 14365C13.pgs 7/12/07 3:57 PM Page 557 19. Triangle RST is inscribed in a circle and mSTX mTRX mRSX b. a. c. Inscribed Angles and Their Measures 557 mRSX 5 mSTX 5 mTRX. Find: d. mR e. mS f. mT Applying Skills 20. In circle O, LM and RS intersect at P. a. Prove that LPR SPM. b. If LP 15 cm, RP 12 cm, and SP 10 cm,... |
Quadrilateral ABCD is inscribed in circle O, and. Prove that ABCD is not a parallelogram. CDX to ABX is not congruent 13-4 TANGENTS AND SECANTS In the diagram, line p has no points in common with the circle. Line m has one point in common with the circle. Line m is said to be tangent to the circle. Line k has two poin... |
that states that at a given point on a circle, one and only one tangent can be drawn. Therefore, our assumption is false and its negation must be true. Line m is perpendicular to OP. We can state Theorems 13.10a and 13.10b as a biconditional. Theorem 13.10 A line is tangent to a circle if and only if it is perpendicul... |
the segments proportional. Tangents and Secants 561 g AB is tangent to circle O at A and circle O at B. A line tangent to a Line circle is perpendicular to a radius drawn to the point of tangency. Since perpendicular lines intersect to form right angles and all right angle are congruent, OAC OBC. Also, OCA OCB because... |
Q > PR OP. The following corollaries are also true. Corollary 13.11a If two tangents are drawn to a circle from an external point, then the line segment from the center of the circle to the external point bisects the angle formed by the tangents. Given g PQ tangent to circle O at Q and g PR tangent to circle O at R. Pr... |
and CE 5, find the perimeter of ABC. Solution Tangent segments drawn to a circle from an exter- nal point are congruent. AD AF 6 BD BE 7 CF CE 5 Therefore AB 5 AD 1 BD 5 6 1 7 5 13 BC 5 BE 1 CE 5 7 1 5 5 12 CA 5 CF 1 AF 5 5 1 6 5 11 Perimeter 5 AB 1 BC 1 CA 5 13 1 12 1 11 5 36 Answer 14365C13.pgs 7/12/07 3:57 PM Page ... |
find the lengths of the sides of the triangle and show that the triangle is a right triangle. A F D PQ In 5–11, T and R. is tangent to circle O at P, SQ is tangent to circle O at S, and OQ intersects circle O at C E B P 5. If OP 15 and PQ 20, find: a. OQ b. SQ c. TQ 6. If OQ 25 and PQ 24, find: a. OP b. RT c. RQ 7. If... |
:57 PM Page 566 566 Geometry of the Circle Applying Skills 15. Prove Corollary 13.11a, “If two tangents are drawn to a circle from an external point, then the line segment from the center of the circle to the external point bisects the angle formed by the tangents.” 16. Prove Corollary 13.11b, “If two tangents are draw... |
07 3:57 PM Page 567 Angles Formed by Tangents, Chords, and Secants 567 Hands-On Activity Consider any regular polygon. Construct the angle bisectors of each interior angle. Since the interior angles are all congruent, the angles formed are all congruent. Since the sides of the regular polygon are all congruent, congrue... |
a chord, and ADC is a right angle because it is an angle inscribed in a semicircle, and ACD is the complement of CAD., BAC is a right angle, and DAB is the Also, complement of CAD. Therefore, since complements of the same angle are congruent, ACD DAB. We can conclude that since mACD =, then mDAB = 2mADX CA'AB 1 1. 2mA... |
S Q R O T P 14365C13.pgs 7/12/07 3:57 PM Page 569 Angles Formed by Tangents, Chords, and Secants 569 mRQP mP mSRQ mP mSRQ mRQP mP mP 2mRQX 2mRTX 2(mRQX 2 mRTX) 1 1 1 Two Intersecting Secants In the diagram, g PTR that intersects the circle at R and T, and is a secant to circle O g PQS is a secant to circle O that inte... |
b. e. mRQP mSQX c. f. mP S Solution Let mQRX 3x, mRSX 5x, and mSQX 7x. 3x 1 5x 1 7x 5 360 15x 5 360 x 5 24 mQRX 5 3x a. mRSX 5 5x b. 5 3(24) 5 72 5 5(24) 5 120 c. mSQX 5 7x 7(24) 5 168 d. m/QRS 5 1 5 1 2mSQX 2(168) e. m/RQP 5 1 5 1 2mQRX 2(72) f. 2(mSQX 2 mQRX ) m/P 5 1 5 1 2(168 2 72) 5 84 5 36 5 48 Answers a. 72° b.... |
/1 5 1 2mABX 2 1 A B D C m/1 5 1 m/2 5 1 2(mABX 1 mCDX) 2(mABX 1 mCDX) (Continued) 14365C13.pgs 7/12/07 3:57 PM Page 572 572 Geometry of the Circle SUMMARY (Continued) Type of Angle Degree Measure Example Formed by Tangents and Secants The measure of an angle formed by a tangent and a secant, two secants, or two tangen... |
find 70 and 9. If 10. If 11. If mQTX mQTX mQRX mQRX mQRX 14. If mP 30 and 13. If 12. If 60 and mP 35, find mQRX 120, find mP. mQTX. mQTX. mQTX. 120, find Q P R T g RP and g QP intersect at P and S is on major arc QRX. 160, find mP. 80, find mP. 260, find mP. 210, find mP. 16. If 15. If In 15–20, tangents mRQX mRQX mRS... |
mCAB c. mACB f. mPAC Applying Skills 29. Tangent g PC intersects circle O at C, chord g AB CP, diameter COD intersects AB at E, and diameter AOF is extended to P. a. Prove that OPC OAE. mADX b. If mOAE 30, find, and mP. mCFX, mFBX, mBDX, mACX, A g ABC 30. Tangent intersects circle O at B, › ‹ AFOD secant intersects th... |
AE EB 5. (AE)(EB) (CE)(ED) 4. The lengths of the corresponding sides of similar triangles are in proportion. 5. In a proportion, the product of the means is equal to the product of the extremes. Segments Formed by a Tangent Intersecting a Secant Do similar relationships exist for tangent segments and secant segments? ... |
Note: This relationship could also have been proved by showing that ABE ADC. 14365C13.pgs 7/12/07 3:57 PM Page 577 Measures of Tangent Segments, Chords, and Secant Segments 577 We can state this as a theorem: Theorem 13.17 If two secant segments are drawn to a circle from an external point, then the product of the len... |
OE is the distance from the center of the circle to the chord. OE 20 DE 5 OD 1 OE 5 25 1 20 5 45 CE 5 OC 2 OE 5 25 2 20 5 5 B 25 c m 5 cm C E A 20 cm O 25 cm D A diameter perpendicular to a chord bisects the chord. Therefore, AE EB. Let AE EB x. (AE)(EB) 5 (DE)(CE) (x)(x) 5 (45)(5) x2 5 225 x 5 15 (Use the positive sq... |
In 3–14, chords AB and intersect at E. CD 3. If CE 12, ED 2, and AE 3, find EB. 4. If CE 16, ED 3, and AE 8, find EB. 5. If AE 20, EB 5, and CE 10, find ED. 6. If AE 14, EB 3, and ED 6, find CE. 7. If CE 10, ED 4, and AE 5, find EB. 8. If CE 56, ED 14, and AE EB, find EB. 9. If CE 12, ED 2, and AE is 2 more than EB, f... |
in the Coordinate Plane 581 28. If AB 4, BC 12, and AD DE, find AE. 29. If AB 2, BC 7, and DE 3, find AD and AE. 30. If AB 6, BC 8, and DE 5, find AD and AE. 31. In a circle, diameter AB is extended through B to P and tangent segment PC is drawn. If BP 6 and PC 9, what is the measure of the diameter of the circle? 13-... |
), (2, 9), and (2, 1) are each 5 units from (2, 4) and are therefore points on the circle. Let P(x, y) be any other point on the circle. From P, draw a vertical line and from C, a horizontal line. (3, 4) y (2, 9) P(x, y) C (2, 4) Q(x, 4) 1 O 1 (2, 1) (7, 4) x 14365C13.pgs 7/12/07 3:57 PM Page 582 582 Geometry of the Ci... |
center of the circle, C, is the midpoint of the diameter. Recall that the coordinates of the midpoint of the segment whose endpoints are (a, b) and (c, d) are 23) 2 B A. The coordinates of C are, 21 1 (21) 2 B 5 (1, 1). y O 1 1 C A x B The length of the radius is the distance from C to any point on the circle. The dis... |
the circle is centered at (3, 2). Shift these endpoints using the translation (7, 0) → (7 3, 0 2) (4, 2) (7, 0) → (7 3, 0 2) (10, 2) T3, 22 : y O 1 1 (0, 0) (3, 2) x METHOD 2 Since the center of the circle is (3, 2), the y-coordinates of the endpoints are both 2. Substitute y 2 into the equation and solve for x: (x 2 ... |
equation to x2 y2 r2. r2 5 50 r 5 6 50 " Since a length is always positive, r 5. Answer r 5 6 r 5 65 " 25 2 " 2 " 2 " Exercises Writing About Mathematics 1. Cabel said that for every circle in the coordinate plane, there is always a diameter that is a vertical line segment and one that is a horizontal line segment. Do... |
5 18 5(x 2 1)2 1 5(y 2 1)2 5 245 28. 29. Point C(2, 3) is the center of a circle and A(3, 9) is a point on the circle. Write an equa- tion of the circle. 30. Does the point (4, 4) lie on the circle whose center is at the origin and whose radius is Justify your answer. 32 "? 31. Is x2 4x 4 y2 2y 1 25 the equation of a ... |
vertices A(1, 3), B(5, 1), and C(5, 3). Find the equation of the circle. Justify your answer algebraically. Circles in the Coordinate Plane 587 A(1, 3) y B(5, 1) C(5, 3) 1 O 1 x 35. Bill Bekebrede wants to build a circular pond in his garden. The garden is in the shape of an equilateral triangle. The length of the alt... |
12/07 3:57 PM Page 588 588 Geometry of the Circle 13-8 TANGENTS AND SECANTS IN THE COORDINATE PLANE Tangents in the Coordinate Plane The circle with center at the origin and radius 5 is shown on the graph. Let l be a line tangent to the circle at A(3, 4). Therefore, since a tangent is perpendicular to the radius drawn ... |
5 0 x2 2 2x 2 48 5 0 (x 8)(x 6 26 26) y 5 8 y (–6, 8) 1 1O x (8, –6) The common solutions are (8, 6) and (6, 8). The line intersects the circle in two points and is therefore a secant. In the diagram, the circle is drawn with its center at the origin and radius 10. The line y 2 x is drawn with a y-intercept of 2 and a... |
5). EXAMPLE 2 The line x + y 2 intersects the circle x2 y2 100 at A(8, 6) and B(6, 8). The line y 10 is tangent to the circle at C(0, 10). a. Find the coordinates of P, the point of intersection of the secant x + y 2 and the tangent y 10. b. Show that PC 2 (PA)(PB). 14365C13.pgs 7/12/07 3:57 PM Page 591 Tangents and S... |
2 y2 10 y 3x 9. x2 y2 25 y = x 1 12. x2 y2 50 x y 10 4. x2 y2 100 x y 14 7. x2 y2 9 y x 3 10. x2 y2 20 x y 6 13. x2 y2 8 x y 4 5. x2 y2 25 x y 7 8. x2 y2 8 x y 11. x2 y2 18 y x 6 14. x2 (y 2)2 4 y x 4 In 15–18, write an equation of the line tangent to the given circle at the given point. 15. x2 y2 9 at (0, 3) 16. x2 y2... |
. a. Show that the points A(1, 7) and B(5, 7) lie on a circle whose radius is 5 and whose cen- ter is at (2, 3). b. What is the distance from the center of the circle to the chord AB? 24. Triangle ABC has vertices A(7, 10), B(2, 2), and C(2, 10). a. Find the coordinates of the points where the circle with equation (x 1... |
ent to each of two circles. • A tangent segment is a segment of a tangent line, one of whose endpoints is the point of tangency. Postulates 13.1 and ABX If point and no other points in common, then ABX m ABX. (Arc Addition Postulate) are two arcs of the same circle having a common endand ABCXBCX BCX BCX ABCX m m 13.2 A... |
ents are drawn to a circle from an external point, then the line segment from the center of the circle to the external point bisects the angle whose vertex is the center of the circle and whose rays are the two radii drawn to the points of tangency. 14365C13.pgs 8/7/07 10:51 AM Page 595 Chapter Summary 595 13.12 The me... |
formed by a tangent and a chord that intersect at the point of tangency is equal to one-half the measure of the intercepted arc. Formed by Two Intersecting Chords The measure of an angle formed by two intersecting chords is equal to one-half the sum of the measures of the arcs intercepted by the angle and its vertical... |
angle of a circle • Arc of a circle • Minor arc • Major arc • Semicircle • Intercepted arc • Degree measure of an arc • Congruent circles • Congruent arcs 13-2 Chord • Diameter • Apothem • Inscribed polygon • Circumscribed circle 13-3 Inscribed angle 14365C13.pgs 7/12/07 3:57 PM Page 598 598 Geometry of the Circle 13-... |
)2 9 A P O B C 9. Two tangents that intersect at P intercept a major arc of 240° on the circle. What is the measure of P? 10. A chord that is 24 centimeters long is 9 centimeters from the center of a circle. What is the measure of the radius of the circle? 14365C13.pgs 8/2/07 6:01 PM Page 599 Review Exercises 599 B O C... |
polygons are not angles that can be constructed using compass and straightedge. For example, a regular polygon with nine sides has angles that measure 140 degrees. Each of the nine isosceles triangles of this polygon has base angles of 70 degrees which cannot be constructed with straightedge and compass. 14365C13.pgs ... |
150 degrees. Bisect this angle to construct the base angle of the isosceles triangles needed to construct a regular dodecagon. CUMULATIVE REVIEW Chapters 1–13 Part I Answer all questions in this part. Each correct answer will receive 2 credits. No partial credit will be allowed. 1. The measure of A is 12 degrees more ... |
) (1, 2) 10. In the diagram, and g AB g AB g CD g CD at E and g EF at F. If intersects mAEF is represented by 3x and mCFE is represented by 2x 20, what is the value of x? (1) 4 (2) 12 (3) 32 (4) 96 A C F E B D 14365C13.pgs 7/12/07 3:57 PM Page 602 602 Geometry of the Circle Part II Answer all questions in this part. Ea... |
07 3:57 PM Page 603 Cumulative Review 603 15. a. Find the coordinates of A, the image of A(5, 2) under the composition R90 + ry 5 x. b. What single transformation is equivalent to R90 + ry 5 x? c. Is R90 + ry 5 x a direct isometry? Justify your answer. ADC 16. In the diagram, D is a point on such that AD : DC 1 : 3, an... |
Chapter 5, we developed procedures to construct the following lines and rays: 1. a line segment congruent to a given line segment 2. an angle congruent to a given angle 3. the bisector of a given line segment 4. the bisector of a given angle 5. a line perpendicular to a given line through a given point on the line 6. ... |
PQ, making PR 2PQ EF 3. On, locate point C at a distance PR from E. 2. Choose any point g AB E on construct the line h EF perpendicular. At E, g AB. to 4. At C, construct g CD perpendicular to g EF and therefore parallel to g.AB Conclusion g CD is parallel to g AB at a distance 2PQ from g.AB 14365C14.pgs 7/10/07 9:59 ... |
Construction 6. Given: ABC Construct: a. a line parallel to g AB at C. b. the median to AC by first constructing the midpoint of AC. c. the median to BC by first constructing the midpoint of BC. C AC and the median to BC intersect at P, can A B d. If the median to the median to point of AB? Justify your answer. AB be ... |
, the length of the radius of a circle. The point of the compass determines a fixed point, point O in the diagram. If the length of the radius remains unchanged, all of the points in the plane that can be drawn by the compass form a circle, and any points that cannot be drawn by the compass do not lie on the circle. Th... |
plane also. EXAMPLE 1 What is the locus of points equidistant from the endpoints of a given line segment? Solution Apply the steps of the procedure for discovering a probable locus. Make a diagram: is the given AB line segment. 2. Decide the condition to be satisfied: P is to be equidistant from A and B. Use a compass... |
12 Locus and Construction Exercises Writing About Mathematics 1. Are all of the points that are equidistant from the endpoints of a line segment that is 8 cen- timeters long 4 centimeters from the endpoints? Explain your answer. 2. What line segment do we measure to find the distance from a point to a line or to a ray?... |
isosceles triangles? 17. Triangle ABC is drawn with a fixed base, AB, and an altitude to AB whose measure is 3 feet. What is the locus of points that can indicate vertex C in all such triangles? 14365C14.pgs 7/10/07 9:59 AM Page 613 Five Fundamental Loci 613 Applying Skills 18. What is the locus of the tip of the hour... |
the angles formed by the intersecting lines. A D Locus s u c o L C B 14365C14.pgs 7/10/07 9:59 AM Page 614 614 Locus and Construction 3. Equidistant from two parallel lines: Find points equidistant from the paral- lel lines g AB and g. CD Locus: The locus of points equidistant from two parallel lines is a third line, ... |
intersecting lines. The locus of points equidistant from intersecting lines is a pair of lines that bisect the angles formed by the g JK g GH are equidistant from given lines. In the figure, g AB g EF (3) The point P at which and g BC. and intersects g EF g GH and the point Q at which intersects are equidistant from g... |
. How many points are equidistant from the two parallel lines and 3 centimeters from P? 12. a. Construct the locus of points equidistant from the endpoints of a line segment. b. Construct the locus of points at a distance AB from M, the midpoint of 1 2(AB). AB c. How many points are equidistant from A and B and at a di... |
d. The locus of points d units from the vertical line x a is the pair of lines x a d and x a d. EXAMPLE 1 What is the equation of the locus of points at a distance of (0, 1)? 20 " units from Solution The locus of points at a fixed distance from a point is the circle with the given point as center and the given distanc... |
ing Skills In 3–8, write an equation of the locus of points at the given distance d from the given point P. 3. d 4, P(0, 0) 6. d 7, P(1, 1) 4. d 1, P(1, 0) 7. d 10, P(3, 1) " 5. d 3, P(0, 2) 8. d 18, P(3, 5) " In 9–12, find the equations of the locus of points at the given distance d from the given line. 9. d 5, x 7 10... |
us of points that are 2 units from (x 4)2 y2 16. b. The locus of points that are 2 units from the y-axis. c. The locus of points that are 2 units from (x 4)2 y2 16 and 2 units from the y-axis. 22. a. The locus of points that are 3 units from (x 1)2 (y 5)2 4. 5 2 3 b. The locus of points that are units from x. 2 3 c. Th... |
) Write an equation of the line through (2, 3) with slope 2 2x 2 4 y 5 2x 2 1 Answer The locus of points equidistant from A(2, 5) and B(6, 1) is the perpendicular. The equation of the locus is y 2x 1. bisector of AB 14365C14.pgs 7/10/07 10:00 AM Page 621 Equidistant Lines in Coordinate Geometry 621 Equidistant from Two... |
. One bisector will have a positive slope and one will have a negative slope. y O 45° B(a, a) A(a, 0) x B(a, a) 14365C14.pgs 7/10/07 10:00 AM Page 622 622 Locus and Construction y O 45° B(a, a) A(a, 0) x B(a, a) Let A(a, 0) be a point on the x-axis and B be a point on the bisector with a positive slope such that is per... |
can be shown that the x-axis bisects the other pair of angles between y mx and y mx. Therefore, the y-axis, together with the x-axis, is the locus of points equidistant from the lines y mx and y mx. A(a, ma) O x The locus of points in the coordinate plane equidistant from two lines with slopes m and m that intersect a... |
the equation of the locus of points equidistant from (3, 1) and (5, 5). b. Prove that the point (2, 6) is equidistant from the points (3, 1) and (5, 5) by showing that it lies on the line whose equation you wrote in a. c. Prove that the point (2, 6) is equidistant from (3, 1) and (5, 5) by using the distance for- mula... |
B and B be points on the lines such that M is is the midpoint of M 0, A. AAr (0, bc) point on and lines. Show that ABM ABM. BBr BBr is perpendicular to both e. Using part d, explain why M is the point at which the line equidistant from the given lines intersects the y-axis. 14-6 POINTS EQUIDISTANT FROM A POINT AND A L... |
given point is and the given line is y 5 21, then the equation of the locus is y x2. Recall that y x2 is the equa4 tion of a parabola whose turning point is the origin and whose axis of symmetry is the y-axis. 0, 1 4 B A Recall that under the translation, the image of 4dy x2 is Th, k 4d(y k) (x h)2. For example, if th... |
2(24) 2(1) 5 5 2 y 5 x2 2 4x 2 1 5 (2)2 2 4(2) 2 1 5 25 c. The axis of symmetry is the vertical line through the turning point, x 2. d. Note that the turning point of the parabola is (2, 5). When the turning point of the parabola y x2 has been moved 2 units to the right and 5 units down, the equation becomes the equat... |
that the equation becomes (y 9) (x (1))2 or y 9 x2 2x 1, which can be written as y x2 2x 8 or y x2 2x 8. 14365C14.pgs 7/10/07 10:00 AM Page 628 628 Locus and Construction EXAMPLE 3 A parabola is equidistant from a given point and a line. How does the turning point of the parabola relate to the given point and line? So... |
2. Amanda said that if the turning point of a parabola is (1, 0), then the x-axis is tangent to the parabola. Do you agree with Amanda? Explain why or why not. Developing Skills In 3–8, find the coordinates of the turning point and the equation of the axis of symmetry of each parabola. 3. y x2 6x 1 6. y x2 2x 5 4. y x... |
fixed points that are the end- points of a segment is the perpendicular bisector of the segment. • The locus of points equidistant from two intersecting lines is a pair of lines that bisect the angles formed by the intersecting lines. • The locus of points equidistant from two parallel lines is a third line, paral- le... |
S on PQ such that PS : SQ 2 : 3. In 3–6, sketch and describe each locus. 3. Equidistant from two points that are 6 centimeters apart. 4. Four centimeters from A and equidistant from A and B, the endpoints of a line segment that is 6 centimeters long. 5. Equidistant from the endpoints of a line segment that is 6 centim... |
Hint: Sketch the locus of points equidistant from each pair of vertices.) Exploration An ellipse is the locus of points such that the sum of the distances from two fixed points F1 and F2 is a constant, k. Use the following procedure to create an ellipse. You will need a piece of string, a piece of thick cardboard, and ... |
AD 3. In triangle ABC, is the altitude to BC then which of the following may be false? (1) AB AC (2) BD CD (3) AB AD (4) mB mC. If D is the midpoint of BC, 4. If two angles of a triangle are congruent and complementary, then the tri- angle is (1) isosceles and right (2) scalene and right (3) isosceles and obtuse (4) s... |
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