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EB = AF ___. FC Construction Triangle Proportionality Theorem Construct a line parallel to a side of a triangle.    Use a straightedge to draw △ABC. Label E on AB. Construct ∠E ≅ ∠B. Label the ̶̶ AC as F. intersection of   EF and ̶̶   EF ǁ BC by the Converse of the Corresponding Angles Postulate. 7- 4 Applyin...
2 Chapter 7 Similarity �� E X A M P L E 3 Art Application ̶̶ AK ǁ An artist used perspective to draw guidelines to help her sketch a row of parallel trees. She then checked the drawing by measuring the distances between the trees. What is LN? ̶̶̶ ̶̶ CM ǁ BL ǁ KL _ =...
�� THINK AND DISCUSS 1. ̶̶ BC. Use what you know about similarity ̶̶ XY ǁ and proportionality to state as many different proportions as possible. 2. GET ORGANIZED Copy and complete the graphic organizer. Draw a figure for each proportionality theorem or coroll...
base. What are LM and MN to the nearest hundredth? �� Find the length of each segment. 13. ̶̶ BC and ̶̶ CD 14. ̶̶ ST and ̶̶ TU = AC_ In the figure,    BC ǁ    DE ǁ    FG. Complete each proportion. 15. AB_ BD 17. DF _ = EG _ CE = AE_ EG = 16. _ DF 18. AF _ AB 20. AB_ AC _...
PR, RT, QS, and SU. b. Use your results from part b to write a proportion relating the segment lengths. Find the length of each segment. 26. ̶̶ EF 27. ̶̶ ST 28. Real Estate A developer is laying out lots along Grant Rd. whose total width is 500 ft. Given the width of each lot along Chavez St., what is the width of eac...
angle bisectors is the same as the ratio of their corresponding sides. 38. Prove the Triangle Angle Bisector Theorem. ̶̶ AD bisects ∠A. Given: In △ABC, = AB _ Prove: BD _ AC DC ̶̶ BX ǁ ̶̶ AD and extend ̶̶ AC to X. Use properties Plan: Draw of parallel lines and the Converse of the Isosceles Triangle Theorem to show th...
Application Eiffel Tower replica in Paris, Texas A student wanted to find the height of a statue of a pineapple in Nambour, Australia. She measured the pineapple’s shadow and her own shadow. The student’s height is 5 ft 4 in. What is the height of the pineapple? Step 1 Convert the measurements to inches. AC = 5 ft 4 i...
5 _ 300 1.5x = 6 (300) 1.5x = 1800 x = 1200 Cross Products Prop. Simplify. Divide both sides by 1.5. The actual distance is 1200 m, or 1.2 km. 2. Find the actual distance between City Hall and El Centro College. E X A M P L E 3 Making a Scale Drawing The Lincoln Memorial in Washington, D.C., is approximately 57 m long ...
Exercises 44 and 45. E X A M P L E 4 Using Ratios to Find Perimeters and Areas Given that △RST ∼ △UVW, find the perimeter P and area A of △UVW. The similarity ratio of △RST to △UVW is 16 __ 20, or 4 __ 5. By the Proportional Perimeters and Areas Theorem, the ratio of the triangles’ perimeters is also 4 __ 5, and the r...
89 Multi-Step A rectangular classroom is 10 m long and 4.6 m wide. Make a scale drawing of the classroom using the following scales. 7. 1 cm : 1 m 8. 1 cm : 2 m 9. 1 cm : 2. Given: rectangle MNPQ ∼ rectangle RSTU p. 490 10. Find the perimeter of rectangle RSTU. 11. Find the area of rectangle RSTU. Independent Practice ...
ABC to the area of △DEF is 16 __ 25. What is the similarity ratio of △ABC to △DEF? 26. The ratio of the area of △ABC to the area of △DEF is 4 __ 81. What is the ratio of the perimeter of △ABC to the perimeter of △DEF? 27. Space Exploration The scale of this model of the space shuttle is 1 ft : 50 ft. In the actual spac...
ft 1 in. tall standing by a statue in Jamestown, North Dakota. Estimate the actual height of the statue by using a ruler to measure her height and the height of the statue in the photo. 33. Math History In A.D. 1076, the mathematician Shen Kua was asked by the emperor of China to produce maps of all Chinese territorie...
AND EXTEND 43. Astronomy The city of Eugene, Oregon, has a scale model of the solar system nearly 6 km long. The model’s scale is 1 km : 1 billion km. a. Earth is 150,000,000 km from the Sun. How many meters apart are Earth and the Sun in the model? b. The diameter of Earth is 12,800 km. What is the diameter, in centi...
, and System Design Q: How do photogrammetrists use math? A: Photogrammetrists use aerial photographs to make detailed maps. To prepare maps, I use computers and perform a lot of scale measures to make sure the maps are accurate. Q: What are your future plans? A: My favorite part of making maps is designing scale drawi...
→ C' (3 ⋅ 3 ) → C' (4.5, 63, 0) → D' (3 ⋅ 3 ) → D' (4.5, 0), 0 ⋅ 3 2 2 Step 2 Plot points A' (0, 0), B' (0, 6), C' (4.5, 6), and D' (4.5, 0). Draw the rectangle. 1. What if…? Draw the border of the original photo after a dilation with scale factor 1 __ 2. 7- 6 Dilations and Similarity in the Coordinate Plane 495 495 �...
) 2 + (4 - 5 AD = √  (-3 - 1) 2 + (1 - 5) 2 AE = √  (5 - 1) 2 + (3 - 5) 2 = √  32 = 4 √  2 = √  20 = 2 √  5 Step 3 Find the similarity ratio. AB _ AD = AC _ AE √  Since AB ___ AD by SAS ∼. = AC ___ AE and ∠A ≅ ∠A by the Reflexive Property, △ABC ∼ △ADE 3. Given: R (-2, 0), S (-3, 1), T (0, 1), U (-5...
√  32 = 4 √  2 BC = √  (3 - 0) 2 + (0 - 1) 2 B'C'= √  (6 - 0) 2 + (0 - 2) 2 = √  10 = √  40 = 2 √  10 AC = √  (3 - 2) 2 + (0 - 3) 2 A'C'= √  (6 - 4) 2 + (0 - 6) 2 = √  10 = √  40 = 2 √  10 Step 4 Find the similarity ratio. A'B' _ AB =, B'C'_ = BC 2 √  10 _ √  10 = 2, A'C'_ = A...
�� 7-6 Exercises Exercises KEYWORD: MG7 7-6 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. A? is a transformation that proportionally reduces or enlarges a figure, ̶̶̶̶ such as the pupil of an ey...
to use on postcards. Sketch the design after a dilation with scale factor 1 __ 2. 11. Given that △UOV ∼ △XOY, find the coordinates of X and the scale factor. 12. Given that △MON ∼ △KOL, find the coordinates of K and the scale factor. 13. Given: D (-1, 3), E (-3, -1), F (3, -1), G (-4, -3), and H (5, -3) Prove: △DEF ∼ ...
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course) 29. A weight lifter must lift at least 250 pounds. There are two 50-pound weights on a bar that weighs 5 pounds. Let w represent the additional weight that must be added to the bar. Find the length of each segment, given that (Lesson 5-2) ̶̶ HF 30. 31. ̶̶ JF ̶̶ DE ≅ ̶̶ FE. 32. ̶̶ CF △SUV ∼ △SRT. Find the lengt...
11 Obj. 3, 8, 10 Use the scale factors given in the above table. Find the relationship between the scale factors of similar figures and their corresponding perimeters. If the relationship is a direct variation, find the constant of variation. 1. regular hexagon with side length 6 2. triangle with side lengths 3, 6, and...
. 6. ̶̶ BC ̶̶ EF 5. 7. 8. A student who is 5 ft 3 in. tall measured her shadow and the shadow cast by a water tower shaped like a golf ball. What is the height of the tower? 7-6 Dilations and Similarity in the Coordinate Plane 9. Given: A (-1, 2), B (-3, -2), C (3, 0), D (-2, 0), and E (1, 1) Prove: △ADE ∼ △ABC 10. Giv...
.... 463 means..................... 455 Complete the sentences below with vocabulary words from the list above. 1. An equation stating that two ratios are equal is called a(n)?. ̶̶̶̶? is a transformation that changes the size of a figure but not its shape. ̶̶̶̶ 2. A(n) 3. In the proportion u _ v = x _ y, the 4. A(n)? c...
.B, G.11.A, G.11.B E X A M P L E EXERCISES ■ Determine whether △ABC and △DEF are similar. If so, write the similarity ratio and a similarity statement. Determine whether the polygons are similar. If so, write the similarity ratio and a similarity statement. 16. rectangles JKLM and PQRS 17. △TUV and △WXY It is given tha...
�� 7-4 Applying Properties of Similar Triangles (pp. 481–487) TEKS G.2.A, G.3.B, G.5.B, G.9.B, G.11.A, G.11.B EXERCISES Find each length. 21. CE 22. ST Verify that the given segments are parallel. 23. ̶̶ KL and ̶̶̶ MN 24. ̶̶ AB and ̶̶ CD 25. Find SU and SV. 26. Find the length of the third side of △ABC. 27. ...
�� 7-5 Using Proportional Relationships (pp. 488–494) TEKS G.1.B, G.5.A, G.11.A, G.11.B, G.11.D E X A M P L E EXERCISES ■ Use the dimensions in the diagram to find the height h of the tower. A student who is 5 ft 5 in. tall measured his shadow and...
4) Prove: △JKL ∼ △JMN 32. Given that △AOB ∼ △COD, find the coordinates of B and the scale factor. Use the Distance Formula to find the side lengths. AC = 2 √  5, AE = 3 √  5 AB = 2 √  10, AD = 3 √  10 = 2 _. 3 Therefore AB _ AD = AC _ AE Since corresponding sides are proportional and ∠A ≅ ∠A by the Reflexive Proper...
�� 11. Given: A (6, 5), B (3, 4), C (6, 3), D (-3, 2), and E (6, -1) Prove: △ABC ∼ △ADE 12. A quilter designed this patch for a quilt but needs a larger version for a different project. Draw the quilt patch after a dilation with scale factor 3 __ 2. 508 508 Chapter 7 Similarity ��...
.2-foot shadow. How tall is the girl? (A) 3 feet 8 inches (B) 4 feet 6 inches (C) 4 feet 7 inches (D) 4 feet 8 inches (E) 5 feet 6 inches 5. What polygon is similar to every other polygon of the same name? (A) Triangle (B) Parallelogram (C) Rectangle (D) Square (E) Trapezoid College Entrance Exam Practice 509 509 ��...
this scale to determine the run of the slope. 3. Write the ratio that represents the slope of m. 4. Anna selected choice B as her answer. Is she correct? If not, what do you think she did wrong? Item B Gridded Response If ABCD ∼ MNOP and AC is 6, what is AB? 5. Examine the figures. Do you think longer or shorter than ...
be sufficient to prove that ABCD is a parallelogram? ̶̶ AB ǁ ̶̶ AC ǁ ̶̶ CD ̶̶ BD ∠CAB ≅ ∠CDB E is the midpoint of ̶̶̶ AD. 7. If   AC is parallel to   BD and m∠1 + m∠2 = 140°, what is the measure of ∠3? 20° 40° 50° 70° 8. If of ̶̶ AC is twice as long as ̶̶ DC? ̶̶ AB, what is the length 2.5 centimeters 3.75 centim...
°, and (x + 27) °? 18. ∠TUV in the diagram below is an obtuse angle. Isosceles acute triangle Isosceles right triangle Scalene acute triangle Scalene obtuse triangle Use the diagram for Items 10 and 11. 10. Which of these points is the orthocenter of △FGH? F G H J Write an inequality showing the range of possible meas...
in meters. six possible similarity statements. a. What is the probability that △ABC ∼ △XYZ is correct? b. If △ABC and △XYZ are isosceles, what is the probability that △ABC ∼ △XYZ? c. If △ABC and △XYZ are equilateral, what is the probability that △ABC ∼ △XYZ? Explain. Extended Response 22. a. Given: △SRT ∼ △VUW and ̶̶ ...
- 3) 12. -2 (y + 5) = -1 14. 2 = -1 (z + 4) Solve Proportions Solve each proportion. 15. 4_ y = 6_ 18 16. 5_ 8 = x_ 32 17. m_ 9 = 8_ 12 18. y_ 4 = 9_ y Rounding and Estimation Round each decimal to the indicated place value. 19. 13.118; hundredth 20. 37.91; tenth 21. 15.992; tenth 22. 173.05; whole number Right Triang...
meaningful problems … G.7.A Dimensionality and the geometry of location* use … two- ★ ★ ★ ★ dimensional coordinate systems … G.8.C Congruence and the geometry of size* … use the ★ ★ ★ Pythagorean Theorem G.10.A Congruence and geometry of size* use congruence transformations to … justify properties of geometric figures...
need answered during class. Right Triangles and Trigonometry 517 517 �� 8-1 Similarity in Right Triangles TEKS G.11.C Similarity and the geometry of shape: develop, apply, and justify triangle similarity relationships, such as right triangle ratios.... Objectives Use geometric mean to find segment lengths in rig...
proportion a __ x = x __ the same number, and that number is the geometric mean of the extremes. The geometric mean of two positive numbers is the positive square root of their product. So the geometric mean of a and b is the positive number x such that x = √  ab, or x 2 = ab.. In this case, the means of the proport...
� z 2 = (12) (2) = 24 z = √  24 = 2 √  6 x is the geometric mean of 2 and 10. Find the positive square root. y is the geometric mean of 12 and 10. Find the positive square root. z is the geometric mean of 12 and 2. Find the positive square root. 3. Find u, v, and w. E X A M P L E 4 Measurement Application To estimat...
necessary, give the answer in simplest radical form. 5. 2 and 50 8. 9 and 12 6. 4 and 16 9. 16 and 25 7. 1 _ 2 and 8 10. 7 and 11 Find x, y, and z. p. 520 11. 12. 13. 520 14. Measurement To estimate the length of the USS Constitution in Boston harbor, a student locates points T and U as shown. What is RS to the neares...
An investment returns 3% one year and 10% the next year. The average rate of return is the geometric mean of the two annual rates. What is the average rate of return for this investment to the nearest tenth of a percent? 41. /////ERROR ANALYSIS///// Two students were asked to find EF. Which solution is incorrect? Expl...
36 square meters 39 square meters 78 square meters 49. Which expression represents the length of ̶̶ RS? √  y + 1) CHALLENGE AND EXTEND 50. Algebra An 8-inch-long altitude of a right triangle divides the hypotenuse into two segments. One segment is 4 times as long as the other. What are the lengths of the segments o...
�� AC with common  AB and 2 Construct point D on  AC. Construct a line  AB. Label the through D perpendicular to intersection of the perpendicular line and  AB as E. 3 Measure ∠A. Measure DE, AE, and AD, the side lengths of △AED. 4 Calculate the ratios DE _ AD, AE _ AD, and DE _. AE Try This 1. Drag D alon...
DEF ∼△XYZ, and BC ___. These are trigonometric AC ratios. A trigonometric ratio is a ratio of two sides of a right triangle. = EF ___ DF = YZ ___ XZ Trigonometric Ratios DEFINITION The sine of an angle is the ratio of the length of the leg opposite the angle to the length of the hypotenuse. The cosine of an angle is th...
 3 _ 2s = √  3 _ 2 The sine of an ∠ is opp. leg _ hyp.. 2. Use a special right triangle to write tan 45° as a fraction. E X A M P L E 3 Calculating Trigonometric Ratios Use your calculator to find each trigonometric ratio. Round to the nearest hundredth. A cos 76° B sin 8° C tan 82° Be sure your calculator is in deg...
= MP Write a trigonometric ratio. Substitute the given values. Multiply both sides by 8.7. MP ≈ 2.98 cm Simplify the expression. C YZ YZ is the hypotenuse. You are given XZ, which is adjacent to the given angle, ∠Z. Since the adjacent side and hypotenuse are involved, use a cosine ratio. cos Z = adj. leg _ hyp. cos 38...
ine ratio to find AB. 3. GET ORGANIZED Copy and complete the graphic organizer. In each cell, write the meaning of each abbreviation and draw a diagram for each. 528 528 Chapter 8 Right Triangles and Trigonometry ge07sec08l02002aABCAB12��34��...
a special right triangle to write each trigonometric ratio as a fraction. 28. tan 60° 29. sin 30° 30. cos 45° Use your calculator to find each trigonometric ratio. Round to the nearest hundredth. 31. tan 51° 34. tan 14° 32. sin 80° 35. sin 55° 33. cos 77° 36. cos 48° 8-2 Trigonometric Ratios 529 529 ��...
AD is perpendicular to the hill, which makes an angle of 28° with a horizontal line. To the nearest inch, how long should this guy wire be? ̶̶ AC be? 530 530 Chapter 8 Right Triangles and Trigonometry �� 53. Find the ...
c. b. Use your results from part a to prove the identity (sin A) 2 + (cos A) 2 = 1. c. Write About It Why do you think this identity is called a Pythagorean identity? Find the perimeter and area of each triangle. Round to the nearest hundredth. 62. 64. 63. 65. 66. Critical Thinking Draw △ABC with ∠C a right angle. Wri...
5x + 10 80. f (x) = x 2 - 4x + 2 Identify the property that justifies each statement. (Lesson 2-5) 81. 82. ̶̶ AB ≅ ̶̶ AB ≅ ̶̶ CD, and ̶̶ AB ̶̶ CD ≅ ̶̶ DE. So ̶̶ AB ≅ ̶̶ DE. 83. If ∠JKM ≅ ∠MLK, then ∠MLK ≅ ∠JKM. Find the geometric mean of each pair of numbers. (Lesson 8-1) 84. 3 and 27 85. 6 and 24 86. 8 and 32 532 532 ...
�� 8-3 Solving Right Triangles TEKS G.11.C Similarity and the geometry of shape: develop, apply, and justify triangle similarity relationships, such as... trigonometric ratios.... Objective Use trigonometric ratios to find angle measures in right triangles and to solve real-world problems. Also G.5.D,...
x = m∠A. If cos A = x, then cos -1 x = m∠A. If tan A = x, then tan -1 x = m∠A. The expression sin -1 x is read “the inverse sine of x.” It does not mean 1 ____. You can think of sin -1 x as “the angle whose sine is x.” sin x 534 534 Chapter 8 Right Triangles and Trigonometry �� E X A ...
≈ 56°. sin A = 5 _ AC, so AC = 5 _. sin A 5 __ ≈ 9.01 ⎤ ⎡ tan -1 ( 5 _ ) ⎥ ⎢ sin ⎦ ⎣ AC ≈ 7.5 3. Find the unknown measures. Round lengths to the nearest hundredth and angle measures to the nearest degree. Solving Right Triangles Rounding can really make a difference! To find AC, I used the Pythagorean Theorem and got ...
- 39° ≈ 51° The acute  of a rt. △ are comp. 4. The coordinates of the vertices of △RST are R (-3, 5), S (4, 5), and T (4, -2). Find the side lengths to the nearest hundredth and the angle measures to the nearest degree. E X A M P L E 5 Travel Application San Francisco’s Lombard Street is known as one of “the crookede...
1) 10. sin -1 (0.5) 8. cos -1 ( 1 _ ) 11. sin -1 (0.61) 3 9. cos -1 ( 5 _ ) 12. tan -1 (0.09. 535 Multi-Step Find the unknown measures. Round lengths to the nearest hundredth and angle measures to the nearest degree. 13. 14. 15. 536 Multi-Step For each triangle, find the side lengths to the nearest hundredth and the an...
. 35. Multi-Step For each triangle, find the side lengths to the nearest hundredth and the angle measures to the nearest degree. 36. A (2, 0), B (2, -5), C (1, -5) 37. M (3, 2), N (3, -2), P (-1, -2) 38. Building For maximum accessibility, a wheelchair ramp should have a slope between 1 __ 16 and 1 __ 20. What is the r...
. The side lengths of a right triangle are given below. Find the measures of the acute angles in the triangle. Round to the nearest degree. 48. 3, 4, 5 49. 5, 12, 13 50. 8, 15, 17 51. What if…? A right triangle has leg lengths of 28 and 45 inches. Suppose the length of the longer leg doubles. What happens to the measur...
A student claims that you must know the three side lengths of a right triangle before you can use trigonometric ratios to find the measures of the acute angles. Do you agree? Why or why not? ̶̶ DC is an altitude of right △ABC. Use trigonometric ratios to find the missing lengths in the figure. Then use these lengths t...
of this altitude.) ̶̶ AC. Use trigonometric SPIRAL REVIEW The graph shows the amount of rainfall in a city for the first five months of the year. Determine whether each statement is true or false. (Previous course) 77. It rained more in April than it did in January, February, and March combined. 78. The average monthl...
. In the figure, ∠ABD is called the line angle. In order to choose the correct weight of the cable for the guy wire, the crew needs to know the measure of the line angle. Find m∠ABD to the nearest degree. 4. To the nearest degree, what is the measure of the angle formed by the pole and the guy wire? 5. What is the perc...
a horizontal line and a line of sight to a point above the line. In the diagram, ∠1 is the angle of elevation from the tower T to the plane P. An angle of depression is the angle formed by a horizontal line and a line of sight to a point below the line. ∠2 is the angle of depression from the plane to the tower. Since ...
foot. E X A M P L E 3 Finding Distance by Using Angle of Depression A forest ranger in a 90-foot observation tower sees a fire. The angle of depression to the fire is 7°. What is the horizontal distance between the tower and the fire? Round to the nearest foot. Draw a sketch to represent the given information. Let T r...
1.9 km So the two towers are about 1.9 km apart. 4. A pilot flying at an altitude of 12,000 ft sights two airports directly in front of him. The angle of depression to one airport is 78°, and the angle of depression to the second airport is 19°. What is the distance between the two airports? Round to the nearest foot....
TEKS TEKS TAKS TAKS Skills Practice p. S19 Application Practice p. S35 PRACTICE AND PROBLEM SOLVING Classify each angle as an angle of elevation or angle of depression. 10. ∠1 11. ∠2 12. ∠3 13. ∠4 14. Geology To measure the height of a rock formation, a surveyor places her transit 100 m from its base and focuses the t...
3, and m∠4. 23. Critical Thinking Describe a situation in which the angle of depression to an object is decreasing. 24. An observer in a hot-air balloon sights a building that is 50 m from the balloon’s launch point. The balloon has risen 165 m. What is the angle of depression from the balloon to the building? Round to...
ALLENGE AND EXTEND 31. Susan and Jorge stand 38 m apart. From Susan’s position, the angle of elevation to the top of Big Ben is 65°. From Jorge’s position, the angle of elevation to the top of Big Ben is 49.5°. To the nearest meter, how tall is Big Ben? �� 32. A plane is flying at a constant altitude of 14,...
Using Trigonometry A clinometer is a surveying tool that is used to measure angles of elevation and angles of depression. In this lab, you will make a simple clinometer and use it to find indirect measurements. Choose a tall object, such as a flagpole or tree, whose height you will measure. TEKS G.11.C Similarity and ...
, engineers have proposed many solutions for lessening the tilt of the Leaning Tower of Pisa. The tower does not form a right angle with the ground, so the engineers have to work with triangles that are not right triangles. In this lesson, you will learn to solve any triangle. To do so, you will need to calculate trigo...
given values. DF sin 105° = 18 sin 32° Cross Products Property In a proportion with three parts, you can use any of the two parts together. DF = 18 sin 32° _ ≈ 9.9 sin 105° B m∠S sin T _ RS sin 75° _ 7 = sin S _ RT = sin S _ 5 sin S = 5 sin 75° _ m∠S ≈ sin -1 ( 5 sin 75° _ 7 7 ) ≈ 44° Divide both sides by sin 105°. La...
2 = RS 2 + RT 2 - 2 (RS) (RT) cos R Law of 9 2 = 4 2 + 7 2 -2 (4 ) (7 ) cos R 81 = 65 - 56 cos R 16 = -56 cos R cos R = - 16 _ 56 m∠R = cos -1 (- 16 _ 56 ) ≈ 107° Cosines Substitute the given values. Simplify. Subtract 65 from both sides. Solve for cos R. Use the inverse cosine function to find m∠R. Find each measure....
using a cable attached from the top of the tower to a point 31 m from the base. How long would this cable be, and what angle would it make with the ground? Round the length to the nearest tenth and the angle measure to the nearest degree. THINK AND DISCUSS 1. Tell what additional information, if any, is needed to find...
angle measures to the nearest degree. 26. m∠C 27. PR 28. JL 29. EF 30. m∠J 31. m∠X 8-5 Law of Sines and Law of Cosines 555 555 �� Surveying Many modern surveys are done with GPS (Global Po...
other given side. In this case, there are two possible triangles. Find two possible values for m∠C to the nearest degree. (Hint: The inverse sine function on your calculator gives you only acute angle measures. Consider this angle and its supplement.) 556 556 Chapter 8 Right Triangles and Trigonometry 6 km4 km3 kmSeco...
altitude. ̶̶ AB into segments of lengths x and y. By the Pythagorean Theorem, It divides a 2 = a. to get c. expression for b 2 to get d. Therefore a 2 = b 2 + c 2 - 2bc cos A by f.?, and b. ̶̶̶̶?. Rearrange the terms to get - 2cx. Substitute the ̶̶̶̶? = h 2 + x 2. Substitute y = c - x into the first equation ̶̶̶̶?. Fr...
have radii of 2 cm, 3 cm, and 4 cm. The centers of the disks form △ABC. Find m∠ACB to the nearest degree. 63. Line ℓ passes through points (-1, 1) and (1, 3). Line m passes through points (-1, 1) and (3, 2). Find the measure of the acute angle formed by ℓ and m to the nearest degree. 64. Navigation The port of Bonner ...
think of a vector as a directed line segment. The vector below may be named  AB or  v. Vocabulary vector component form magnitude direction equal vectors parallel vectors resultant vector Also G.1.B, G.7.C, G.11.C A vector can also be named using component form. The component form 〈x, y〉 of a vector lists the hor...
X A M P L E 2 Finding the Magnitude of a Vector Draw the vector 〈4, -2〉 on a coordinate plane. Find its magnitude to the nearest tenth. Step 1 Draw the vector on a coordinate plane. Use the origin as the initial point. Then (4, -2) is the terminal point. Step 2 Find the magnitude. Use the Distance Formula. ⎜〈4, -2〉⎟ =...
initial point and terminal point. Two vectors are parallel vectors if they have the same direction or if they have opposite directions. They may have different magnitudes. For example, vectors are always parallel vectors.  x. Equal  w ǁ  BA  AB ≠ Note that since these vectors do not have the same direction. ...
E 5 Sports Application A kayaker leaves shore at a bearing of N 55° E and paddles at a constant speed of 3 mi/h. There is a 1 mi/h current moving due east. What are the kayak’s actual speed and direction? Round the speed to the nearest tenth and the direction to the nearest degree. Step 1 Sketch vectors for the kayake...
_ 5. What if…? Suppose the kayaker in Example 5 instead paddles at 4 mi/h at a bearing of N 20° E. What are the kayak’s actual speed and direction? Round the speed to the nearest tenth and the direction to the nearest degree. 562 562 Chapter 8 Right Triangles and Trigonometry ��...
, 3〉. Identify each of the following. p. 561 13. equal vectors in diagram 1 14. parallel vectors in diagram 1 15. equal vectors in diagram 2 16. parallel vectors in diagram 2 8-6 Vectors 563 563 ��. 562 17. Recreation To reach a campsite, a hiker...
plane’s actual speed and direction? Round the speed to the nearest tenth and the direction to the nearest degree. Find each vector sum. 32. 〈1, 2〉 + 〈0, 6〉 34. 〈0, 1〉 + 〈7, 0〉 33. 〈-3, 4〉 + 〈5, -2〉 35. 〈8, 3〉 + 〈-2, -1〉 36. Critical Thinking Is vector addition commutative? That is, is  u +  v equal to  v +  u? Use...
and a long pole must be used to close it. a. Carla holds the pole at a 45° angle to the floor and applies 10 lb of force to the upper edge of the window. Find the vertical component of the vector representing the force on the window. Round to the nearest tenth. b. Taneka also applies 10 lb of force to close the window...
of vectors. Then find the magnitude and direction of the resultant vector. Round the magnitude to the nearest tenth and the direction to the nearest degree. 53.  u = 〈1, 2〉,  v = 〈4.8, -3.1〉  v = 〈2.5, -1〉  u = 〈-2, 7〉, 54. 55.  u = 〈6, 0〉,  v = 〈-2, 4〉 56.  u = 〈-1.2, 8〉,  v = 〈5.2, -2.1〉 57. Math History In ...
� u  v  w  z 61. The vector 〈7, 9〉 represents the velocity of a helicopter. What is the direction of this vector to the nearest degree? 38° 52° 128° 142° 62. A canoe sets out on a course given by the vector 〈5, 11〉. What is the length of the canoe’s course to the nearest unit? 6 8 12 16 63. Gridded Response  AB ...
�JLM is 6 cm 2, find each measure. (Lesson 7-5) 72. the perimeter of △NPS 73. the area of △NPS Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. (Lesson 8-5) 74. BC 75. m∠B 76. m∠C 8-6 Vectors 567 567 �� SECTION 8B Applying Tri...
6 m long. What is the height of the building to the nearest tenth of a meter? 8-5 Law of Sines and Law of Cosines Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. 3. m∠A 4. GH 5. XZ 6. UV 7. m∠F 8. QS 8-6 Vectors Draw each vector on a coordinate plane. Find its magnitude ...
345° Reference angle: 15° Angle measure: 435° Reference angle: 75° E X A M P L E 1 Finding Reference Angles Sketch each angle on the coordinate plane. Find the measure of its reference angle. A 102° B 236° Reference angle: 180° - 102° = 78° Reference angle: 236° - 180° = 56° Sketch each angle on the coordinate plane. ...
y) is in Quadrant IV, its y-coordinate is negative. So the coordinates of P are ( √  2 ___ 2, - √  2 ___ 2 ). Remember that tan θ = sin θ _. So tan 315° = sin 315° _ = cos θ cos 315° - √  2 ___ 2 _ √  2 ___ 2 = -1. Find each trigonometric ratio. 2a. cos 240° 2b. sin 135° EXTENSION Exercises Exercises Sketch each a...
....... 561 vector...................... 559 direction................... 560 resultant vector............. 561 Complete the sentences below with vocabulary words from the list above. 1. The? of a vector gives the horizontal and vertical change from the initial point ̶̶̶̶ to the terminal point. 2. Two vectors with the ...
8) (11) z 2 = 88 z = √  88 = 2 √  22 z is the geometric mean of 8 and 11. 572 572 Chapter 8 Right Triangles and Trigonometry �� 8-2 Trigonometric Ratios (pp. 525–532) TEKS G.5.B, G.5.D, G.8.C, G.11.B, G.11.C E X A M P L E S EXERCISES Find each length. Round to the nearest hu...
�� 8-4 Angles of Elevation and Depression (pp. 544–549) TEKS G.5.D, G.11.C E X A M P L E S EXERCISES Classify each angle as an angle of elevation or angle of depression. ■ A pilot in a plane spots a forest fire on the ground at an angle of depression of 71°...
�� Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. ■ HJ Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. 26. EF Use the Law of Cosines. 27. m∠Q HJ 2 = GH 2 + G J 2 - 2 (GH) (GJ) ...
〈0.48, 0.13〉. The current is 〈1, 0〉. Susan’s actual speed is the magnitude of the resultant vector, 〈1.48, 0.13〉. ⎜〈1.48, 0.13〉⎟ = Her direction is tan -1 ( 0.13 _ √  (1.48) 2 + (0.13) 2 ≈ 1.5 mi/h ) ≈ 5°, or N 85° E. 1.48 Draw each vector on a coordinate plane. Find its magnitude to the nearest tenth. 31. 〈-5,...
. Round lengths to the nearest tenth and angle measures to the nearest degree. 12. m∠B 13. RS 14. m∠M Draw each vector on a coordinate plane. Find its magnitude to the nearest tenth. 15. 〈1, 3〉 16. 〈-4, 1〉 17. 〈2, -3〉 Draw each vector on a coordinate plane. Find the direction of the vector to the nearest degree. 18. Th...
about 6 minutes to complete. 1. Let P be the acute angle formed by the line -x + 4y = 12 and the x-axis. What is the approximate measure of ∠P? (A) 14° (B) 18° (C) 72° (D) 76° (E) 85° 2. In right triangle DEF, DE = 15, EF = 36, and DF = 39. What is the cosine of ∠F? (A) 5 _ 12 (B) 12 _ 5 (C) 5 _ 13 (D) 12 _ 13 (E) 13 ...
extra minute spent checking your answers can result in a better test score. Item C Multiple Choice In △QRS, what is the ̶̶ SQ to the nearest tenth of a measure of centimeter? Read each test item and answer the questions that follow. Item A Gridded Response A cell phone tower casts a shadow that is 121 ft long when the...
24% 3.89% 10.50% 7. Describe how to use estimation to show that choices F and J are unreasonable. 8. To find the answer, a student uses the equation x 2 = (2.625) (4). Which compatible numbers should the student use to quickly check the answer? TAKS Tackler 579 579 �� KEYWORD: MG7 TestPrep CUMULAT...
right triangle. m∠A = 20°, m∠B = 90°, AC = 8, and AB = 3. Which expression can be used to find BC? 3 _ tan 70° 8 _ sin 20° 8 tan 20° 3 cos 70° 3. A slide at a park is 25 ft long, and the top of the slide is 10 ft above the ground. What is the approximate measure of the angle the slide makes with the ground? 21.8° 23.6...

EB = AF ___. FC Construction Triangle Proportionality Theorem Construct a line parallel to a side of a triangle.    Use a straightedge to draw △ABC. Label E on AB. Construct ∠E ≅ ∠B. Label the ̶̶ AC as F. intersection of   EF and ̶̶   EF ǁ BC by the Converse of the Corresponding Angles Postulate. 7- 4 Applyin...

2 Chapter 7 Similarity �� E X A M P L E 3 Art Application ̶̶ AK ǁ An artist used perspective to draw guidelines to help her sketch a row of parallel trees. She then checked the drawing by measuring the distances between the trees. What is LN? ̶̶̶ ̶̶ CM ǁ BL ǁ KL _ =...

�� THINK AND DISCUSS 1. ̶̶ BC. Use what you know about similarity ̶̶ XY ǁ and proportionality to state as many different proportions as possible. 2. GET ORGANIZED Copy and complete the graphic organizer. Draw a figure for each proportionality theorem or coroll...

base. What are LM and MN to the nearest hundredth? �� Find the length of each segment. 13. ̶̶ BC and ̶̶ CD 14. ̶̶ ST and ̶̶ TU = AC_ In the figure,    BC ǁ    DE ǁ    FG. Complete each proportion. 15. AB_ BD 17. DF _ = EG _ CE = AE_ EG = 16. _ DF 18. AF _ AB 20. AB_ AC _...

PR, RT, QS, and SU. b. Use your results from part b to write a proportion relating the segment lengths. Find the length of each segment. 26. ̶̶ EF 27. ̶̶ ST 28. Real Estate A developer is laying out lots along Grant Rd. whose total width is 500 ft. Given the width of each lot along Chavez St., what is the width of eac...

angle bisectors is the same as the ratio of their corresponding sides. 38. Prove the Triangle Angle Bisector Theorem. ̶̶ AD bisects ∠A. Given: In △ABC, = AB _ Prove: BD _ AC DC ̶̶ BX ǁ ̶̶ AD and extend ̶̶ AC to X. Use properties Plan: Draw of parallel lines and the Converse of the Isosceles Triangle Theorem to show th...

Application Eiffel Tower replica in Paris, Texas A student wanted to find the height of a statue of a pineapple in Nambour, Australia. She measured the pineapple’s shadow and her own shadow. The student’s height is 5 ft 4 in. What is the height of the pineapple? Step 1 Convert the measurements to inches. AC = 5 ft 4 i...

5 _ 300 1.5x = 6 (300) 1.5x = 1800 x = 1200 Cross Products Prop. Simplify. Divide both sides by 1.5. The actual distance is 1200 m, or 1.2 km. 2. Find the actual distance between City Hall and El Centro College. E X A M P L E 3 Making a Scale Drawing The Lincoln Memorial in Washington, D.C., is approximately 57 m long ...

Exercises 44 and 45. E X A M P L E 4 Using Ratios to Find Perimeters and Areas Given that △RST ∼ △UVW, find the perimeter P and area A of △UVW. The similarity ratio of △RST to △UVW is 16 __ 20, or 4 __ 5. By the Proportional Perimeters and Areas Theorem, the ratio of the triangles’ perimeters is also 4 __ 5, and the r...

89 Multi-Step A rectangular classroom is 10 m long and 4.6 m wide. Make a scale drawing of the classroom using the following scales. 7. 1 cm : 1 m 8. 1 cm : 2 m 9. 1 cm : 2. Given: rectangle MNPQ ∼ rectangle RSTU p. 490 10. Find the perimeter of rectangle RSTU. 11. Find the area of rectangle RSTU. Independent Practice ...

ABC to the area of △DEF is 16 __ 25. What is the similarity ratio of △ABC to △DEF? 26. The ratio of the area of △ABC to the area of △DEF is 4 __ 81. What is the ratio of the perimeter of △ABC to the perimeter of △DEF? 27. Space Exploration The scale of this model of the space shuttle is 1 ft : 50 ft. In the actual spac...

ft 1 in. tall standing by a statue in Jamestown, North Dakota. Estimate the actual height of the statue by using a ruler to measure her height and the height of the statue in the photo. 33. Math History In A.D. 1076, the mathematician Shen Kua was asked by the emperor of China to produce maps of all Chinese territorie...

AND EXTEND 43. Astronomy The city of Eugene, Oregon, has a scale model of the solar system nearly 6 km long. The model’s scale is 1 km : 1 billion km. a. Earth is 150,000,000 km from the Sun. How many meters apart are Earth and the Sun in the model? b. The diameter of Earth is 12,800 km. What is the diameter, in centi...

, and System Design Q: How do photogrammetrists use math? A: Photogrammetrists use aerial photographs to make detailed maps. To prepare maps, I use computers and perform a lot of scale measures to make sure the maps are accurate. Q: What are your future plans? A: My favorite part of making maps is designing scale drawi...

→ C' (3 ⋅ 3 ) → C' (4.5, 63, 0) → D' (3 ⋅ 3 ) → D' (4.5, 0), 0 ⋅ 3 2 2 Step 2 Plot points A' (0, 0), B' (0, 6), C' (4.5, 6), and D' (4.5, 0). Draw the rectangle. 1. What if…? Draw the border of the original photo after a dilation with scale factor 1 __ 2. 7- 6 Dilations and Similarity in the Coordinate Plane 495 495 �...

) 2 + (4 - 5 AD = √  (-3 - 1) 2 + (1 - 5) 2 AE = √  (5 - 1) 2 + (3 - 5) 2 = √  32 = 4 √  2 = √  20 = 2 √  5 Step 3 Find the similarity ratio. AB _ AD = AC _ AE √  Since AB ___ AD by SAS ∼. = AC ___ AE and ∠A ≅ ∠A by the Reflexive Property, △ABC ∼ △ADE 3. Given: R (-2, 0), S (-3, 1), T (0, 1), U (-5...

√  32 = 4 √  2 BC = √  (3 - 0) 2 + (0 - 1) 2 B'C'= √  (6 - 0) 2 + (0 - 2) 2 = √  10 = √  40 = 2 √  10 AC = √  (3 - 2) 2 + (0 - 3) 2 A'C'= √  (6 - 4) 2 + (0 - 6) 2 = √  10 = √  40 = 2 √  10 Step 4 Find the similarity ratio. A'B' _ AB =, B'C'_ = BC 2 √  10 _ √  10 = 2, A'C'_ = A...

�� 7-6 Exercises Exercises KEYWORD: MG7 7-6 KEYWORD: MG7 Parent GUIDED PRACTICE Vocabulary Apply the vocabulary from this lesson to answer each question. 1. A? is a transformation that proportionally reduces or enlarges a figure, ̶̶̶̶ such as the pupil of an ey...

to use on postcards. Sketch the design after a dilation with scale factor 1 __ 2. 11. Given that △UOV ∼ △XOY, find the coordinates of X and the scale factor. 12. Given that △MON ∼ △KOL, find the coordinates of K and the scale factor. 13. Given: D (-1, 3), E (-3, -1), F (3, -1), G (-4, -3), and H (5, -3) Prove: △DEF ∼ ...

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course) 29. A weight lifter must lift at least 250 pounds. There are two 50-pound weights on a bar that weighs 5 pounds. Let w represent the additional weight that must be added to the bar. Find the length of each segment, given that (Lesson 5-2) ̶̶ HF 30. 31. ̶̶ JF ̶̶ DE ≅ ̶̶ FE. 32. ̶̶ CF △SUV ∼ △SRT. Find the lengt...

11 Obj. 3, 8, 10 Use the scale factors given in the above table. Find the relationship between the scale factors of similar figures and their corresponding perimeters. If the relationship is a direct variation, find the constant of variation. 1. regular hexagon with side length 6 2. triangle with side lengths 3, 6, and...

. 6. ̶̶ BC ̶̶ EF 5. 7. 8. A student who is 5 ft 3 in. tall measured her shadow and the shadow cast by a water tower shaped like a golf ball. What is the height of the tower? 7-6 Dilations and Similarity in the Coordinate Plane 9. Given: A (-1, 2), B (-3, -2), C (3, 0), D (-2, 0), and E (1, 1) Prove: △ADE ∼ △ABC 10. Giv...

.... 463 means..................... 455 Complete the sentences below with vocabulary words from the list above. 1. An equation stating that two ratios are equal is called a(n)?. ̶̶̶̶? is a transformation that changes the size of a figure but not its shape. ̶̶̶̶ 2. A(n) 3. In the proportion u _ v = x _ y, the 4. A(n)? c...

.B, G.11.A, G.11.B E X A M P L E EXERCISES ■ Determine whether △ABC and △DEF are similar. If so, write the similarity ratio and a similarity statement. Determine whether the polygons are similar. If so, write the similarity ratio and a similarity statement. 16. rectangles JKLM and PQRS 17. △TUV and △WXY It is given tha...

�� 7-4 Applying Properties of Similar Triangles (pp. 481–487) TEKS G.2.A, G.3.B, G.5.B, G.9.B, G.11.A, G.11.B EXERCISES Find each length. 21. CE 22. ST Verify that the given segments are parallel. 23. ̶̶ KL and ̶̶̶ MN 24. ̶̶ AB and ̶̶ CD 25. Find SU and SV. 26. Find the length of the third side of △ABC. 27. ...

�� 7-5 Using Proportional Relationships (pp. 488–494) TEKS G.1.B, G.5.A, G.11.A, G.11.B, G.11.D E X A M P L E EXERCISES ■ Use the dimensions in the diagram to find the height h of the tower. A student who is 5 ft 5 in. tall measured his shadow and...

4) Prove: △JKL ∼ △JMN 32. Given that △AOB ∼ △COD, find the coordinates of B and the scale factor. Use the Distance Formula to find the side lengths. AC = 2 √  5, AE = 3 √  5 AB = 2 √  10, AD = 3 √  10 = 2 _. 3 Therefore AB _ AD = AC _ AE Since corresponding sides are proportional and ∠A ≅ ∠A by the Reflexive Proper...

�� 11. Given: A (6, 5), B (3, 4), C (6, 3), D (-3, 2), and E (6, -1) Prove: △ABC ∼ △ADE 12. A quilter designed this patch for a quilt but needs a larger version for a different project. Draw the quilt patch after a dilation with scale factor 3 __ 2. 508 508 Chapter 7 Similarity ��...

.2-foot shadow. How tall is the girl? (A) 3 feet 8 inches (B) 4 feet 6 inches (C) 4 feet 7 inches (D) 4 feet 8 inches (E) 5 feet 6 inches 5. What polygon is similar to every other polygon of the same name? (A) Triangle (B) Parallelogram (C) Rectangle (D) Square (E) Trapezoid College Entrance Exam Practice 509 509 ��...

this scale to determine the run of the slope. 3. Write the ratio that represents the slope of m. 4. Anna selected choice B as her answer. Is she correct? If not, what do you think she did wrong? Item B Gridded Response If ABCD ∼ MNOP and AC is 6, what is AB? 5. Examine the figures. Do you think longer or shorter than ...

be sufficient to prove that ABCD is a parallelogram? ̶̶ AB ǁ ̶̶ AC ǁ ̶̶ CD ̶̶ BD ∠CAB ≅ ∠CDB E is the midpoint of ̶̶̶ AD. 7. If   AC is parallel to   BD and m∠1 + m∠2 = 140°, what is the measure of ∠3? 20° 40° 50° 70° 8. If of ̶̶ AC is twice as long as ̶̶ DC? ̶̶ AB, what is the length 2.5 centimeters 3.75 centim...

°, and (x + 27) °? 18. ∠TUV in the diagram below is an obtuse angle. Isosceles acute triangle Isosceles right triangle Scalene acute triangle Scalene obtuse triangle Use the diagram for Items 10 and 11. 10. Which of these points is the orthocenter of △FGH? F G H J Write an inequality showing the range of possible meas...

in meters. six possible similarity statements. a. What is the probability that △ABC ∼ △XYZ is correct? b. If △ABC and △XYZ are isosceles, what is the probability that △ABC ∼ △XYZ? c. If △ABC and △XYZ are equilateral, what is the probability that △ABC ∼ △XYZ? Explain. Extended Response 22. a. Given: △SRT ∼ △VUW and ̶̶ ...

- 3) 12. -2 (y + 5) = -1 14. 2 = -1 (z + 4) Solve Proportions Solve each proportion. 15. 4_ y = 6_ 18 16. 5_ 8 = x_ 32 17. m_ 9 = 8_ 12 18. y_ 4 = 9_ y Rounding and Estimation Round each decimal to the indicated place value. 19. 13.118; hundredth 20. 37.91; tenth 21. 15.992; tenth 22. 173.05; whole number Right Triang...

meaningful problems … G.7.A Dimensionality and the geometry of location* use … two- ★ ★ ★ ★ dimensional coordinate systems … G.8.C Congruence and the geometry of size* … use the ★ ★ ★ Pythagorean Theorem G.10.A Congruence and geometry of size* use congruence transformations to … justify properties of geometric figures...

need answered during class. Right Triangles and Trigonometry 517 517 �� 8-1 Similarity in Right Triangles TEKS G.11.C Similarity and the geometry of shape: develop, apply, and justify triangle similarity relationships, such as right triangle ratios.... Objectives Use geometric mean to find segment lengths in rig...

proportion a __ x = x __ the same number, and that number is the geometric mean of the extremes. The geometric mean of two positive numbers is the positive square root of their product. So the geometric mean of a and b is the positive number x such that x = √  ab, or x 2 = ab.. In this case, the means of the proport...

� z 2 = (12) (2) = 24 z = √  24 = 2 √  6 x is the geometric mean of 2 and 10. Find the positive square root. y is the geometric mean of 12 and 10. Find the positive square root. z is the geometric mean of 12 and 2. Find the positive square root. 3. Find u, v, and w. E X A M P L E 4 Measurement Application To estimat...

necessary, give the answer in simplest radical form. 5. 2 and 50 8. 9 and 12 6. 4 and 16 9. 16 and 25 7. 1 _ 2 and 8 10. 7 and 11 Find x, y, and z. p. 520 11. 12. 13. 520 14. Measurement To estimate the length of the USS Constitution in Boston harbor, a student locates points T and U as shown. What is RS to the neares...

An investment returns 3% one year and 10% the next year. The average rate of return is the geometric mean of the two annual rates. What is the average rate of return for this investment to the nearest tenth of a percent? 41. /////ERROR ANALYSIS///// Two students were asked to find EF. Which solution is incorrect? Expl...

36 square meters 39 square meters 78 square meters 49. Which expression represents the length of ̶̶ RS? √  y + 1) CHALLENGE AND EXTEND 50. Algebra An 8-inch-long altitude of a right triangle divides the hypotenuse into two segments. One segment is 4 times as long as the other. What are the lengths of the segments o...

�� AC with common  AB and 2 Construct point D on  AC. Construct a line  AB. Label the through D perpendicular to intersection of the perpendicular line and  AB as E. 3 Measure ∠A. Measure DE, AE, and AD, the side lengths of △AED. 4 Calculate the ratios DE _ AD, AE _ AD, and DE _. AE Try This 1. Drag D alon...

DEF ∼△XYZ, and BC ___. These are trigonometric AC ratios. A trigonometric ratio is a ratio of two sides of a right triangle. = EF ___ DF = YZ ___ XZ Trigonometric Ratios DEFINITION The sine of an angle is the ratio of the length of the leg opposite the angle to the length of the hypotenuse. The cosine of an angle is th...

 3 _ 2s = √  3 _ 2 The sine of an ∠ is opp. leg _ hyp.. 2. Use a special right triangle to write tan 45° as a fraction. E X A M P L E 3 Calculating Trigonometric Ratios Use your calculator to find each trigonometric ratio. Round to the nearest hundredth. A cos 76° B sin 8° C tan 82° Be sure your calculator is in deg...

= MP Write a trigonometric ratio. Substitute the given values. Multiply both sides by 8.7. MP ≈ 2.98 cm Simplify the expression. C YZ YZ is the hypotenuse. You are given XZ, which is adjacent to the given angle, ∠Z. Since the adjacent side and hypotenuse are involved, use a cosine ratio. cos Z = adj. leg _ hyp. cos 38...

ine ratio to find AB. 3. GET ORGANIZED Copy and complete the graphic organizer. In each cell, write the meaning of each abbreviation and draw a diagram for each. 528 528 Chapter 8 Right Triangles and Trigonometry ge07sec08l02002aABCAB12��34��...

a special right triangle to write each trigonometric ratio as a fraction. 28. tan 60° 29. sin 30° 30. cos 45° Use your calculator to find each trigonometric ratio. Round to the nearest hundredth. 31. tan 51° 34. tan 14° 32. sin 80° 35. sin 55° 33. cos 77° 36. cos 48° 8-2 Trigonometric Ratios 529 529 ��...

AD is perpendicular to the hill, which makes an angle of 28° with a horizontal line. To the nearest inch, how long should this guy wire be? ̶̶ AC be? 530 530 Chapter 8 Right Triangles and Trigonometry �� 53. Find the ...

c. b. Use your results from part a to prove the identity (sin A) 2 + (cos A) 2 = 1. c. Write About It Why do you think this identity is called a Pythagorean identity? Find the perimeter and area of each triangle. Round to the nearest hundredth. 62. 64. 63. 65. 66. Critical Thinking Draw △ABC with ∠C a right angle. Wri...

5x + 10 80. f (x) = x 2 - 4x + 2 Identify the property that justifies each statement. (Lesson 2-5) 81. 82. ̶̶ AB ≅ ̶̶ AB ≅ ̶̶ CD, and ̶̶ AB ̶̶ CD ≅ ̶̶ DE. So ̶̶ AB ≅ ̶̶ DE. 83. If ∠JKM ≅ ∠MLK, then ∠MLK ≅ ∠JKM. Find the geometric mean of each pair of numbers. (Lesson 8-1) 84. 3 and 27 85. 6 and 24 86. 8 and 32 532 532 ...

�� 8-3 Solving Right Triangles TEKS G.11.C Similarity and the geometry of shape: develop, apply, and justify triangle similarity relationships, such as... trigonometric ratios.... Objective Use trigonometric ratios to find angle measures in right triangles and to solve real-world problems. Also G.5.D,...

x = m∠A. If cos A = x, then cos -1 x = m∠A. If tan A = x, then tan -1 x = m∠A. The expression sin -1 x is read “the inverse sine of x.” It does not mean 1 ____. You can think of sin -1 x as “the angle whose sine is x.” sin x 534 534 Chapter 8 Right Triangles and Trigonometry �� E X A ...

≈ 56°. sin A = 5 _ AC, so AC = 5 _. sin A 5 __ ≈ 9.01 ⎤ ⎡ tan -1 ( 5 _ ) ⎥ ⎢ sin ⎦ ⎣ AC ≈ 7.5 3. Find the unknown measures. Round lengths to the nearest hundredth and angle measures to the nearest degree. Solving Right Triangles Rounding can really make a difference! To find AC, I used the Pythagorean Theorem and got ...

- 39° ≈ 51° The acute  of a rt. △ are comp. 4. The coordinates of the vertices of △RST are R (-3, 5), S (4, 5), and T (4, -2). Find the side lengths to the nearest hundredth and the angle measures to the nearest degree. E X A M P L E 5 Travel Application San Francisco’s Lombard Street is known as one of “the crookede...

1) 10. sin -1 (0.5) 8. cos -1 ( 1 _ ) 11. sin -1 (0.61) 3 9. cos -1 ( 5 _ ) 12. tan -1 (0.09. 535 Multi-Step Find the unknown measures. Round lengths to the nearest hundredth and angle measures to the nearest degree. 13. 14. 15. 536 Multi-Step For each triangle, find the side lengths to the nearest hundredth and the an...

. 35. Multi-Step For each triangle, find the side lengths to the nearest hundredth and the angle measures to the nearest degree. 36. A (2, 0), B (2, -5), C (1, -5) 37. M (3, 2), N (3, -2), P (-1, -2) 38. Building For maximum accessibility, a wheelchair ramp should have a slope between 1 __ 16 and 1 __ 20. What is the r...

. The side lengths of a right triangle are given below. Find the measures of the acute angles in the triangle. Round to the nearest degree. 48. 3, 4, 5 49. 5, 12, 13 50. 8, 15, 17 51. What if…? A right triangle has leg lengths of 28 and 45 inches. Suppose the length of the longer leg doubles. What happens to the measur...

A student claims that you must know the three side lengths of a right triangle before you can use trigonometric ratios to find the measures of the acute angles. Do you agree? Why or why not? ̶̶ DC is an altitude of right △ABC. Use trigonometric ratios to find the missing lengths in the figure. Then use these lengths t...

of this altitude.) ̶̶ AC. Use trigonometric SPIRAL REVIEW The graph shows the amount of rainfall in a city for the first five months of the year. Determine whether each statement is true or false. (Previous course) 77. It rained more in April than it did in January, February, and March combined. 78. The average monthl...

. In the figure, ∠ABD is called the line angle. In order to choose the correct weight of the cable for the guy wire, the crew needs to know the measure of the line angle. Find m∠ABD to the nearest degree. 4. To the nearest degree, what is the measure of the angle formed by the pole and the guy wire? 5. What is the perc...

a horizontal line and a line of sight to a point above the line. In the diagram, ∠1 is the angle of elevation from the tower T to the plane P. An angle of depression is the angle formed by a horizontal line and a line of sight to a point below the line. ∠2 is the angle of depression from the plane to the tower. Since ...

foot. E X A M P L E 3 Finding Distance by Using Angle of Depression A forest ranger in a 90-foot observation tower sees a fire. The angle of depression to the fire is 7°. What is the horizontal distance between the tower and the fire? Round to the nearest foot. Draw a sketch to represent the given information. Let T r...

1.9 km So the two towers are about 1.9 km apart. 4. A pilot flying at an altitude of 12,000 ft sights two airports directly in front of him. The angle of depression to one airport is 78°, and the angle of depression to the second airport is 19°. What is the distance between the two airports? Round to the nearest foot....

TEKS TEKS TAKS TAKS Skills Practice p. S19 Application Practice p. S35 PRACTICE AND PROBLEM SOLVING Classify each angle as an angle of elevation or angle of depression. 10. ∠1 11. ∠2 12. ∠3 13. ∠4 14. Geology To measure the height of a rock formation, a surveyor places her transit 100 m from its base and focuses the t...

3, and m∠4. 23. Critical Thinking Describe a situation in which the angle of depression to an object is decreasing. 24. An observer in a hot-air balloon sights a building that is 50 m from the balloon’s launch point. The balloon has risen 165 m. What is the angle of depression from the balloon to the building? Round to...

ALLENGE AND EXTEND 31. Susan and Jorge stand 38 m apart. From Susan’s position, the angle of elevation to the top of Big Ben is 65°. From Jorge’s position, the angle of elevation to the top of Big Ben is 49.5°. To the nearest meter, how tall is Big Ben? �� 32. A plane is flying at a constant altitude of 14,...

Using Trigonometry A clinometer is a surveying tool that is used to measure angles of elevation and angles of depression. In this lab, you will make a simple clinometer and use it to find indirect measurements. Choose a tall object, such as a flagpole or tree, whose height you will measure. TEKS G.11.C Similarity and ...

, engineers have proposed many solutions for lessening the tilt of the Leaning Tower of Pisa. The tower does not form a right angle with the ground, so the engineers have to work with triangles that are not right triangles. In this lesson, you will learn to solve any triangle. To do so, you will need to calculate trigo...

given values. DF sin 105° = 18 sin 32° Cross Products Property In a proportion with three parts, you can use any of the two parts together. DF = 18 sin 32° _ ≈ 9.9 sin 105° B m∠S sin T _ RS sin 75° _ 7 = sin S _ RT = sin S _ 5 sin S = 5 sin 75° _ m∠S ≈ sin -1 ( 5 sin 75° _ 7 7 ) ≈ 44° Divide both sides by sin 105°. La...

2 = RS 2 + RT 2 - 2 (RS) (RT) cos R Law of 9 2 = 4 2 + 7 2 -2 (4 ) (7 ) cos R 81 = 65 - 56 cos R 16 = -56 cos R cos R = - 16 _ 56 m∠R = cos -1 (- 16 _ 56 ) ≈ 107° Cosines Substitute the given values. Simplify. Subtract 65 from both sides. Solve for cos R. Use the inverse cosine function to find m∠R. Find each measure....

using a cable attached from the top of the tower to a point 31 m from the base. How long would this cable be, and what angle would it make with the ground? Round the length to the nearest tenth and the angle measure to the nearest degree. THINK AND DISCUSS 1. Tell what additional information, if any, is needed to find...

angle measures to the nearest degree. 26. m∠C 27. PR 28. JL 29. EF 30. m∠J 31. m∠X 8-5 Law of Sines and Law of Cosines 555 555 �� Surveying Many modern surveys are done with GPS (Global Po...

other given side. In this case, there are two possible triangles. Find two possible values for m∠C to the nearest degree. (Hint: The inverse sine function on your calculator gives you only acute angle measures. Consider this angle and its supplement.) 556 556 Chapter 8 Right Triangles and Trigonometry 6 km4 km3 kmSeco...

altitude. ̶̶ AB into segments of lengths x and y. By the Pythagorean Theorem, It divides a 2 = a. to get c. expression for b 2 to get d. Therefore a 2 = b 2 + c 2 - 2bc cos A by f.?, and b. ̶̶̶̶?. Rearrange the terms to get - 2cx. Substitute the ̶̶̶̶? = h 2 + x 2. Substitute y = c - x into the first equation ̶̶̶̶?. Fr...

have radii of 2 cm, 3 cm, and 4 cm. The centers of the disks form △ABC. Find m∠ACB to the nearest degree. 63. Line ℓ passes through points (-1, 1) and (1, 3). Line m passes through points (-1, 1) and (3, 2). Find the measure of the acute angle formed by ℓ and m to the nearest degree. 64. Navigation The port of Bonner ...

think of a vector as a directed line segment. The vector below may be named  AB or  v. Vocabulary vector component form magnitude direction equal vectors parallel vectors resultant vector Also G.1.B, G.7.C, G.11.C A vector can also be named using component form. The component form 〈x, y〉 of a vector lists the hor...

X A M P L E 2 Finding the Magnitude of a Vector Draw the vector 〈4, -2〉 on a coordinate plane. Find its magnitude to the nearest tenth. Step 1 Draw the vector on a coordinate plane. Use the origin as the initial point. Then (4, -2) is the terminal point. Step 2 Find the magnitude. Use the Distance Formula. ⎜〈4, -2〉⎟ =...

initial point and terminal point. Two vectors are parallel vectors if they have the same direction or if they have opposite directions. They may have different magnitudes. For example, vectors are always parallel vectors.  x. Equal  w ǁ  BA  AB ≠ Note that since these vectors do not have the same direction. ...

E 5 Sports Application A kayaker leaves shore at a bearing of N 55° E and paddles at a constant speed of 3 mi/h. There is a 1 mi/h current moving due east. What are the kayak’s actual speed and direction? Round the speed to the nearest tenth and the direction to the nearest degree. Step 1 Sketch vectors for the kayake...

_ 5. What if…? Suppose the kayaker in Example 5 instead paddles at 4 mi/h at a bearing of N 20° E. What are the kayak’s actual speed and direction? Round the speed to the nearest tenth and the direction to the nearest degree. 562 562 Chapter 8 Right Triangles and Trigonometry ��...

, 3〉. Identify each of the following. p. 561 13. equal vectors in diagram 1 14. parallel vectors in diagram 1 15. equal vectors in diagram 2 16. parallel vectors in diagram 2 8-6 Vectors 563 563 ��. 562 17. Recreation To reach a campsite, a hiker...

plane’s actual speed and direction? Round the speed to the nearest tenth and the direction to the nearest degree. Find each vector sum. 32. 〈1, 2〉 + 〈0, 6〉 34. 〈0, 1〉 + 〈7, 0〉 33. 〈-3, 4〉 + 〈5, -2〉 35. 〈8, 3〉 + 〈-2, -1〉 36. Critical Thinking Is vector addition commutative? That is, is  u +  v equal to  v +  u? Use...

and a long pole must be used to close it. a. Carla holds the pole at a 45° angle to the floor and applies 10 lb of force to the upper edge of the window. Find the vertical component of the vector representing the force on the window. Round to the nearest tenth. b. Taneka also applies 10 lb of force to close the window...

of vectors. Then find the magnitude and direction of the resultant vector. Round the magnitude to the nearest tenth and the direction to the nearest degree. 53.  u = 〈1, 2〉,  v = 〈4.8, -3.1〉  v = 〈2.5, -1〉  u = 〈-2, 7〉, 54. 55.  u = 〈6, 0〉,  v = 〈-2, 4〉 56.  u = 〈-1.2, 8〉,  v = 〈5.2, -2.1〉 57. Math History In ...

� u  v  w  z 61. The vector 〈7, 9〉 represents the velocity of a helicopter. What is the direction of this vector to the nearest degree? 38° 52° 128° 142° 62. A canoe sets out on a course given by the vector 〈5, 11〉. What is the length of the canoe’s course to the nearest unit? 6 8 12 16 63. Gridded Response  AB ...

�JLM is 6 cm 2, find each measure. (Lesson 7-5) 72. the perimeter of △NPS 73. the area of △NPS Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. (Lesson 8-5) 74. BC 75. m∠B 76. m∠C 8-6 Vectors 567 567 �� SECTION 8B Applying Tri...

6 m long. What is the height of the building to the nearest tenth of a meter? 8-5 Law of Sines and Law of Cosines Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. 3. m∠A 4. GH 5. XZ 6. UV 7. m∠F 8. QS 8-6 Vectors Draw each vector on a coordinate plane. Find its magnitude ...

345° Reference angle: 15° Angle measure: 435° Reference angle: 75° E X A M P L E 1 Finding Reference Angles Sketch each angle on the coordinate plane. Find the measure of its reference angle. A 102° B 236° Reference angle: 180° - 102° = 78° Reference angle: 236° - 180° = 56° Sketch each angle on the coordinate plane. ...

y) is in Quadrant IV, its y-coordinate is negative. So the coordinates of P are ( √  2 ___ 2, - √  2 ___ 2 ). Remember that tan θ = sin θ _. So tan 315° = sin 315° _ = cos θ cos 315° - √  2 ___ 2 _ √  2 ___ 2 = -1. Find each trigonometric ratio. 2a. cos 240° 2b. sin 135° EXTENSION Exercises Exercises Sketch each a...

....... 561 vector...................... 559 direction................... 560 resultant vector............. 561 Complete the sentences below with vocabulary words from the list above. 1. The? of a vector gives the horizontal and vertical change from the initial point ̶̶̶̶ to the terminal point. 2. Two vectors with the ...

8) (11) z 2 = 88 z = √  88 = 2 √  22 z is the geometric mean of 8 and 11. 572 572 Chapter 8 Right Triangles and Trigonometry �� 8-2 Trigonometric Ratios (pp. 525–532) TEKS G.5.B, G.5.D, G.8.C, G.11.B, G.11.C E X A M P L E S EXERCISES Find each length. Round to the nearest hu...

�� 8-4 Angles of Elevation and Depression (pp. 544–549) TEKS G.5.D, G.11.C E X A M P L E S EXERCISES Classify each angle as an angle of elevation or angle of depression. ■ A pilot in a plane spots a forest fire on the ground at an angle of depression of 71°...

�� Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. ■ HJ Find each measure. Round lengths to the nearest tenth and angle measures to the nearest degree. 26. EF Use the Law of Cosines. 27. m∠Q HJ 2 = GH 2 + G J 2 - 2 (GH) (GJ) ...

〈0.48, 0.13〉. The current is 〈1, 0〉. Susan’s actual speed is the magnitude of the resultant vector, 〈1.48, 0.13〉. ⎜〈1.48, 0.13〉⎟ = Her direction is tan -1 ( 0.13 _ √  (1.48) 2 + (0.13) 2 ≈ 1.5 mi/h ) ≈ 5°, or N 85° E. 1.48 Draw each vector on a coordinate plane. Find its magnitude to the nearest tenth. 31. 〈-5,...

. Round lengths to the nearest tenth and angle measures to the nearest degree. 12. m∠B 13. RS 14. m∠M Draw each vector on a coordinate plane. Find its magnitude to the nearest tenth. 15. 〈1, 3〉 16. 〈-4, 1〉 17. 〈2, -3〉 Draw each vector on a coordinate plane. Find the direction of the vector to the nearest degree. 18. Th...

about 6 minutes to complete. 1. Let P be the acute angle formed by the line -x + 4y = 12 and the x-axis. What is the approximate measure of ∠P? (A) 14° (B) 18° (C) 72° (D) 76° (E) 85° 2. In right triangle DEF, DE = 15, EF = 36, and DF = 39. What is the cosine of ∠F? (A) 5 _ 12 (B) 12 _ 5 (C) 5 _ 13 (D) 12 _ 13 (E) 13 ...

extra minute spent checking your answers can result in a better test score. Item C Multiple Choice In △QRS, what is the ̶̶ SQ to the nearest tenth of a measure of centimeter? Read each test item and answer the questions that follow. Item A Gridded Response A cell phone tower casts a shadow that is 121 ft long when the...

24% 3.89% 10.50% 7. Describe how to use estimation to show that choices F and J are unreasonable. 8. To find the answer, a student uses the equation x 2 = (2.625) (4). Which compatible numbers should the student use to quickly check the answer? TAKS Tackler 579 579 �� KEYWORD: MG7 TestPrep CUMULAT...

right triangle. m∠A = 20°, m∠B = 90°, AC = 8, and AB = 3. Which expression can be used to find BC? 3 _ tan 70° 8 _ sin 20° 8 tan 20° 3 cos 70° 3. A slide at a park is 25 ft long, and the top of the slide is 10 ft above the ground. What is the approximate measure of the angle the slide makes with the ground? 21.8° 23.6...