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1 Geometry/Trigonometry Unit 7: Right Triangle Notes Name: Date: Period: # (1) Page 430 #1 15 (2) Page #16 23, 25 27, 29 and 31 (3) Page #1 8, 9 19 odd (4) Page #10 20 Even, 23, and 29 (5) Page 444 #1 14 (6) Page #15 26, (7) Page 451 #1 8 (8) Page #9 20 (9) Page 457 #1 10 (10) Page #11 28 (11) Page 465 #1 18 (12) Page #19 29 Odd, all (13) Page #1 8, 12, 13, 15, 16, Even, 29 32, (14) Page 473 #1 21
2 Geometry Notes 9.1 Exploring Right Triangles ll right triangles are either To prove that two right triangles are similar you can show that of one of the triangles is to of the of the other triangle. Theorem 9.1 (Use 3x5 ard to Explore): If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other. Therefore:. What triangles are similar? Theorem 9.2: In a right triangle, the length of the altitude from the right angle to the hypotenuse is the geometric mean of the lengths of the two segments of the hypotenuse. Theorem 9.3: In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. Each leg of the right triangle is the geometric mean of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg. and
3 Geometry Notes 9.2 The Pythagorean Theorem Theorem 9.4 The Pythagorean Theorem: In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs. set of three positive integers, a, b and c, that satisfy the equation is a Pythagorean Triple. (ex1) 3, 4, 5 (ex2) (ex3) (ex4) 5, 12, 13 (ex5) (ex6) (ex7) 7, 24, 25 (ex8) (ex9) Geometry Notes 9.3 The onverse of the Pythagorean Theorem Theorem 9.5 onverse of the Pythagorean Theorem: If the square of the length of the longest side of a triangle is equal to the sum of the squares of the two shorter sides, then the triangle is a right triangle. Theorem 9.6: If the square of the length of the longest side of a triangle is less than the sum of the squares of the lengths of the two shorter sides, then the triangle is acute. If a 2 + b 2 > c 2 or c 2 < a 2 + b 2, then the triangle is an acute triangle Theorem 9.7: If the square of the length of the longest side of a triangle is greater than the sum of the squares of the lengths of the two shorter sides, then the triangle is obtuse. If a 2 + b 2 < c 2 or c 2 > a 2 + b 2, then the triangle is an obtuse triangle
4 Geometry Notes 9.4 Special Right Triangles times as long as each leg. That is, the side- Theorem 9.8 In a triangle, the hypotenuse is length ratios of leg:leg:hyp are 1:1:. E1. Find the missing values E2. Find the missing values P1. Find the missing values P2. Find the missing values Theorem 9.9 In a triangle, the hypotenuse is twice as long as the shorter leg and the longer leg is as long as the shorter leg. That is, the side-length ratios of short leg : long leg : hyp is 1: :2 E1. Find t he missing values E2. Find the missing values E3. Find the missing values P1. Find the missing values P2. Find the missing values P3. Find the missing values
5 Geometry Notes 9.5 Trigonometric Ratios measurement of triangles From <: From <: a c a c b b Trigonometric Ratio a ratio of the lengths of two sides of a triangle. Sine (sin), osine (cos) and Tangent (tan) are the basic ratios Ex1. Find each trigonometric ratio: (a) (b) sin = cos = (c) tan = (d) sin = (e) cos = (f) tan = Find the missing side or angle and round to the nearest tenth E2. E3. E4. E5.
6 Geometry Notes 9.6 Extended pplication: Solving Right Triangles: Solving a right triangle means to find the measure of of the triangle and of the triangle. In other words all parts. You can solve a right triangle if you know: (1) Two lengths OR (2) One length and one measure. How to find an angle measure of a right triangle given two side measures E1. Sin =.2548 E2. os =.5624 E3. Tan = P1. Sin =.6895 P2. os =.4156 P3. Tan =. How to solve a right triangle given two side lengths E4. P How to solve a right triangle given one side length and an acute angle measure E5. P5. 3 7
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