9-1 Similar Right Triangles (Day 1) 1. Review:

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1 9-1 Similar Right Triangles (Day 1) 1. Review: Given: ACB is right and AB CD Prove: ΔADC ~ ΔACB ~ ΔCDB. Statement Reason 2. In the diagram in #1, suppose AD = 27 and BD = 3. Find CD. (You may find it helps to draw the three triangles separately or write the proportions from the similarity statements first.) The altitude is the Geometric Mean between 3. Refer to the figure below. (a) If AC = 12 and BC = 5, find CD. (Hint: Find AB first.) The product of the hypotenuse and the altitude to the hypotenuse equals Ch 9 Right Triangles 1

2 (b) If AD = 16 and BC = 15, find BD. The length of each leg is the Geometric Mean between (c) If BD = x, AD = x + 7, BC = y, AC = z, and CD = 12, find x, y, and z. (Hint: Find x first.) 4. Given: PQRS is a rectangle. PS is the geometric mean between ST and TR. Prove: PTQ is a right angle. Hint: First show that ΔPST ~ ΔTRQ using SAS~ Ch 9 Right Triangles 2

3 5. In right ΔDEF, the length of the altitude DG to hypotenuse EF is 20. If EF = 58, find all possible values of EG. (DYOD) 6. Find x, y, and z. Express your answer in exact value (i.e. square roots). 7. In the diagram, ABCD is a square with AS = BP = CQ = DR and AP = BQ = CR = DS as shown. Without using the Pythagorean Theorem, explain why PQRS is also a square. Ch 9 Right Triangles 3 End

4 9-2 The Pythagorean Theorem (Day 2) 1. In this problem, you will give an alternate proof of the Pythagorean Theorem. a.) Using similar triangles, show that a 2 = cx and b 2 = cy. b.) Using (a), show that a 2 + b 2 = c Consider isosceles ΔABC with base AB = 12 and AC = Find its area. (Hint: Cut it in half.) 3. In parallelogram ABCD, AB = 10, AC = 12, and BD = 16. Show that ABCD is a rhombus. (Hint: Show that the diagonals are ) Ch 9 Right Triangles 4

5 4. Last night, you showed that if ABCD is a square, then so is PQRS. Using this, you will prove the Pythagorean Theorem. (a) Express the area of square ABCD in terms of a and b. (b) Express the sum of the areas of the four right triangles and square PQRS n terms of a, b, and c. (c) Show that a 2 + b 2 = c 2 by setting the two areas equal. 5. Find the length of the longest stick that can fit in the box shown below. 6. Three integers {a, b, c} form a Pythagorean triple if a 2 + b 2 = c 2. For example, {3, 4, 5} is a Pythagorean triple because = 5 2. (a) Name another Pythagorean triple. (b) Pick any two positive integers m and n with m > n. Compute the following: Numbers chosen: m = n = m 2 n 2 = 2mn = m 2 + n 2 = What can you say about these three numbers?? Ch 9 Right Triangles 5

6 (c) Show algebraically why this method always produces a Pythagorean triple. 7. In ΔJKL, K is a right angle. If JK = x + 2, KL = 2x, and JL = x + 6, find all possible values of x. 8. The length of one leg of a right triangle is 8 more than that of the other leg. If the perimeter of the triangle is 4 times the length of the shorter leg, find the lengths of all three sides. 9. Find the value of h. (Hint: Let BC = x. So AC = ) Ch 9 Right Triangles 6 End

7 9-3 Special Right Triangles (Day 3) 1. Consider the triangle below. a. If AB = 6, find AC and BC. Be sure to simplify the square root. b. If AB = 10, find AC and BC. c. If AB = 2a, find AC and BC in terms of a. 2. Consider the isosceles right triangle below. d. If AC = 3, find BC and AB. e. If AC = 5, find BC and AB. f. If AC = a, find BC and AB in terms of a. Ch 9 Right Triangles 7

8 3. The two diagrams below summarize the results of Problems 1 and 2. Memorize these now so you can quickly solve these special right triangles without relying on the Pythagorean Theorem each time. 4. Consider the triangle below. Lizzy says that x = 9. Ivy claims that x = 3 3. Who is correct? 3 Explain. 5. One angle of a rhombus measures 120. If the length of the longer diagonal is 12, find the length of the shorter diagonal and the perimeter of the rhombus. Ch 9 Right Triangles 8

9 6. In each of the diagrams below, solve for x and y. 7. If the circle shown has radius 10, find the area of the shaded region. Leave your answer in terms of π. Ch 9 Right Triangles 9

10 8. The circle shown below has center R and m PRQ = 120. Find the area of the shaded region. (Hint: 120 is 1/3 of the entire circle.) 9. In quadrilateral QRST, m R = 60, m T = 90, QR = RS, ST = 8, and TQ = 8. Find the length of diagonal RT. (Hint: QRST is a kite.) 10. The solid (3D figure) shown below is composed of four equilateral triangles. A and B are midpoints of the two edges shown. Find AB. Ch 9 Right Triangles 10 End

11 9-4 Trigonometric Ratios (Day 4) After this class, you should know, 1) How to set up an equation using the 3 basic trig. ratios, 2) How to solve an equation for a missing side of a right triangle, 3) How to solve an equation for a missing angle of a right triangle, 4) Why we need trigonometry. 3 Basic Trig. Definitions In a right triangle, the following ratios are defined for acute angles: Sine of = Cosine of = Tangent of = theta is a common variable used in trigonometry for angles. Sin = Cos = Tan = Note: Find a way to memorize these. Soh-Cah-Toa, Oscar Had A Heap Of Apples, etc. Ex. 1 (a) Without using a calculator, find the sine of 60 degrees in the triangle below. sin60 = (b) Now find sin60 in the triangle on the right. sin60 = (c) Again, find sin60 in the triangle below with no given side lengths. Hmm sin60 = Big Questions: Why do these trig. ratios always work? What does this have to do with similar triangles? (d) Find cos60 and tan60. cos60 = tan60 = (exact value) Verify your results with the calculator (make sure you are in degree mode). Ch 9 Right Triangles 11

12 Ex. 2 You are given the triangle below and need to find the values of x and y. (a) Why do we need trigonometry to solve this problem? (b) Are the sides labeled x and y fixed or can they change? Why? (c) Using your trig. ratios and your calculator, find the values of x and y. Approximate your answers to two decimal places. Ex. 3 Consider the triangle below. (a) Are the angles in this triangle fixed or can there be more than one measure for A? (b) Find sina. sin A = (c) Now, to find the measure of angle A, type ( inverse sine also known as arcsine ) Practice: 1. If tan B = 3 4 find C BC = 9 AB = tan C = A B m C = 2. Find the values of x, y and z. Approximate your answers to two decimal places. 40 o 8 y 38 o x z Ch 9 Right Triangles 12

13 3. Find m K given that L is a right angle. 21 L K Without using a calculator, find the exact values of sin 45, cos 45, and tan 45. Then verify your results with the calculator. sin 45 = cos 45 = tan 45 = 5. How are sin 30, cos 30, and tan 30 related to sin 60, cos 60, and tan 60? Explain. 6. Using your calculator, evaluate the following. a. (sin(23 )) 2 + (cos(23 )) 2 b. (sin(47 )) 2 + (cos(47 )) 2 c. (sin(62 )) 2 + (cos(62 )) 2 d. Make a conjecture. 7. Using the diagram below, algebraically prove the following identities. Start with one expression and manipulate it to show that it equals the other expression. Don t start with them being equal!! a. (sin A) 2 + (cos A) 2 = 1 b. tan A = sin A cos A c. sin A = cos(90 A) Ch 9 Right Triangles 13

14 8. In Δ ABC, AB = AC = 13, and BC = 10. (a) Find sin B, cos B, and tan B. sin B = cos B = tan B = (b) Then find the measures of the 3 angles. B C A 9. Diagonals of rectangle ABCD are 18 cm long and intersect at a 34 degree angle. Find the perimeter of ABCD. Approximate your answer to two decimal places. 10. In the cube below, find m CAB. Approximate your answer to two decimal places. End Ch 9 Right Triangles 14

15 9-5 Using Trigonometry (Day 5) Angle of Elevation/Angle of Depression 1. A person at window W, 40 ft above street level, sights points on a building directly across the street. H is chosen so that WH is horizontal. T is directly above H, and B is directly below. If m TWH = 61 and m BWH = 37, how far above street level is T? 2. Find m A to the nearest hundredth of a degree. 3. Suppose sinθ = 2 where θ (theta) is an angle between 0 and a. Without using a calculator, find the exact value of cosθ and tanθ (DYOD) b. Using a calculator, verify your results in a. by first finding θ and then evaluating cosθ and tanθ. c. Verify your results in a. again by evaluating (sinθ) 2 + (cosθ) 2. Ch 9 Right Triangles 15

16 4. The sides of a rectangle have lengths 14 and 4.8. Find the measure of one of the acute angles formed by the diagonals of the rectangle. (DYOD) 5. In the diagram, AB = 15, m A = 42, and m CBD = 33. a. Find the perimeter of ΔABC. b. Find the area of ΔABC. 6. A rhombus has a 70 degree angle. Its shorter diagonal is 122 cm. How long, to the nearest centimeter, is the longer diagonal? 7. In ΔABC, m C = 90, m A = 24, and median CD is 6cm long. (Recall: A median connects a vertex to a midpoint.) a. Show that CD BD b. Find the perimeter of ΔABC. Ch 9 Right Triangles 16 End

17 Ch 9 Review (Day 6) 1. Consider ΔABC. a. Find x. b. Find the area of ΔABC. Express your answer as an exact value. 2. A rectangle is 80 cm long and 20 cm wide. Find, to the nearest degree, the larger angle formed at the intersection of the diagonals. 3. In ΔABC, m B = m C = 72, and BC = 10. a. Find AB to two decimal places. b. Find the length of the bisector of A to BC to two decimal places. Ch 9 Right Triangles 17

18 4. Find x and y. Round your answers to the nearest integer. 5. Find the perimeter of ΔABC. 6. To which of the following does cos20 equal? sin 20 sin 70 cos If sinθ = 3, find the exact value of cosθ and tanθ Algebraically prove: sin A a = sin B b (Hint: Draw the altitude from C.) Ch 9 Right Triangles 18

19 9. Consider the diagram to the right. a. Draw the altitude from C to AB. Label the point where they meet D. b. Find CD c. Find AD and DB d. Find BC e. Find m B f. Find m C 10. In ΔABC, m B = 105, m C = 30, and BC = 8. Find the perimeter of the triangle. Express your answer as an exact value. (Hint: Draw an auxiliary line.) In the diagram below, AB =, m ABG = m CDB = 45, m FBG = 30, and DF = FG. Find 2 the value of x. Express your answer as an exact value. Ch 9 Right Triangles 19

20 12. In the triangle below AB = 12. Find the values of x, y, and z. 13. John is 2m tall. He is at the top of a 42m tall building. When he looks down at an angle of depression of 25, he sees the top of a second building. The base of the second building is 50 meters away from the base of John s building. How tall is the second building? 2m 42m 50m Selected Answers: 1a. x = 2 1b. Area = a b x= 119; y = sin 70 because cosφ = sin(90 φ) 7. cosθ = 3 7 tanθ = or (1) sin A = b x therefore, x = b(sina); sin B = a x therefore, x = a(sinb) (2) b(sina) = a(sinb) sin A sin B (3) = 9b. 4sin40 = c. AD = 4cos40 = 3.064; DB = AB AD = = a b d. BC = (use Pythag Thm) 9e. sin 1 & # B = $! = % " 9f. C = B = x = 12. x = 9; y = 6.301; z = Second building = = m or 6 6 Ch 9 Right Triangles 20

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