13.3 Special Right Triangles
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1 Name lass ate. Special Right Triangles Essential Question: What do you know about the side lengths and the trigonometric ratios in special right triangles? Eplore Investigating an Isosceles Right Triangle Resource Locker iscover relationships that always apply in an isosceles right triangle. The figure shows an isosceles right triangle. Identify the base angles, and use the fact that they are complementary to write an equation relating their measures., ; m + m = 90 Use the Isosceles Triangle Theorem to write a different equation relating the base angle measures. m = m What must the measures of the base angles be? Why? m = m = ; using substitution, m = 90, so m =. Use the Pythagorean Theorem to find the length of the otenuse in terms of the length of each leg,. = + = = Houghton Mifflin Harcourt Publishing ompany Reflect. Is it true that if you know one side length of an isosceles right triangle, then you know all the side lengths? Eplain. Yes; If you know either leg length, then the other leg length is also equal to, and the length of the otenuse is this value multiplied by. If you know the otenuse length, then each leg length is this value divided by.. What if? Suppose you draw the perpendicular from to. Eplain how to find the length of. Since is isosceles, m =, and since is a right triangle, m = 90 - m =. Therefore is an isosceles right triangle, so = or. Module 709 Lesson O NOT EIT--hanges must be made through "File info" orrectionkey=nl-;- ate
2 Eplore Investigating nother Special Right Triangle iscover relationships that always apply in a right triangle formed as half of an equilateral triangle. is an equilateral triangle and is a perpendicular from to. etermine all three angle measures in. m = 90 ; each angle in an equilateral triangle measures, so m =, and therefore m = 90 - m = 0. Eplain why. (Equilateral Triangle Theorem), (all right angles are congruent), and (Refleive Property of ongruence), so by S ongruence. Or, ( equilateral), (Refleive Property of ongruence), and since and are right triangles, by HL ongruence. Let the length of be. What is the length of, and why? From Step,, so (PT) and therefore = =. In, = + = + = (Seg. dd. Post.); since ( equilateral), = =. Using the Pythagorean Theorem, find the length of. () = + = = Reflect. What is the numerical ratio of the side lengths in a right triangle with acute angles that measure 0 and? Eplain. The right triangle is similar to ( Similarity), so its side lengths are in the ratio : :, or : :. 4. Eplain the Error student has drawn a right triangle with a angle and a otenuse of 6. He has labeled the other side lengths as shown. Eplain how you can tell at a glance that he has made an error and how to correct it. Since 6 > 6, leg JK is longer than the otenuse, which is impossible. The side lengths must be in the ratio : :. For this to be true, the length of JK should be. L J 6 6 K Houghton Mifflin Harcourt Publishing ompany Module 70 Lesson
3 Eplain pplying Relationships in Special Right Triangles The right triangles you eplored are sometimes called --90 and triangles. In a --90 triangle, the otenuse is times as long as each leg. In a triangle, the otenuse is twice as long as the shorter leg and the longer leg is times as long as the shorter leg. You can use these relationships to find side lengths in these special types of right triangles. 0 Eample Find the unknown side lengths in each right triangle. Find the unknown side lengths in. The otenuse is times as long as each leg. = = Substitute 0 for. 0 = = Multiply by. 0 = = ivide by. 5 = = 0 In right EF, m = 0 and m E =. The shorter leg measures 5. Find the remaining side lengths. Houghton Mifflin Harcourt Publishing ompany 0 E 5 The otenuse is twice as long as the shorter leg. E = Substitute 5 for EF. E = Simplify. E = The longer leg is times as long as the shorter leg. = F EF Substitute 5 for EF. F = 5 Simplify. F = F EF Module 7 Lesson
4 Your Turn Find the unknown side lengths in each right triangle. 5. L 0 6. Q 4 J K KL = JK JL = JK 4 = JK JL = ( = JK JL = 6 ) P 6 PR QR, so QR = PR = 6 PQ = PR = ( 6 ) = 4 R Eplain Trigonometric Ratios of Special Right Triangles You can use the relationships you found in special right triangles to find trigonometric ratios for the angles, 0, and. Eample For each triangle, find the unknown side lengths and trigonometric ratios for the angles triangle with a leg length of Step Since the lengths of the sides opposite the angles are congruent, they are both. The length of the otenuse is times as long as each leg, so it is ( ), or. Step Use the triangle to find the trigonometric ratios for. Write each ratio as a simplified fraction. opp ngle Sine = adj osine = opp Tangent = adj Houghton Mifflin Harcourt Publishing ompany Module 7 Lesson
5 triangle with a shorter leg of Step The otenuse is twice as long as the shorter leg, so the length of the otenuse is. The longer leg is times as long as the shorter leg, so the length of the longer leg is. Step 0 Use the triangle to complete the table. Write each ratio as a simplified fraction. ngle 0 Sine = opp osine = adj Tangent = opp adj Reflect 7. Write any patterns or relationships you see in the tables in Part and Part as equations. Why do these patterns or relationships make sense? sin 0 = cos = ; sin = cos = ; sin = cos 0 = ; the sine of an angle equals the cosine of its complement. 8. For which acute angle measure θ, is tanθ less than? equal to? greater than? tan θ < for θ < ; tan θ = for θ = ; tan θ > for θ >. Houghton Mifflin Harcourt Publishing ompany Your Turn Find the unknown side lengths and trigonometric ratios for the angles = 0, = = 5 ; =, = 5 sin = cos = 5 0 = tan = Module 7 Lesson
6 Eplain Investigating Pythagorean Triples Pythagorean Triples Pythagorean triple is a set of positive integers a, b, and c that satisfy the equation a + b = c. This means that a, b, and c are the legs and otenuse of a right triangle. Right triangles that have non-integer sides will not form Pythagorean triples. Eamples of Pythagorean triples include, 4, and 5; 5,, and ; 7, 4, and 5; and 8, 5, and 7. c b a Eample Use Pythagorean triples to find side lengths in right triangles. Verify that the side lengths, 4, and 5; 5,, and ; 7, 4, and 5; and 8, 5, and 7 are Pythagorean triples. The numbers in Step are not the only Pythagorean triples. In the following steps you will discover that multiples of known Pythagorean triples are also Pythagorean triples. + 4 = = 5 = = = 69 = = = 65 = = = 89 = 7 In right triangles EF and JKL, a, b, and c form a Pythagorean triple, and k is a positive integer greater than. Eplain how the two triangles are related. J a F b c E ka L kc kb K EF is similar to JKL by the Side-Side-Side (SSS) Triangle Similarity Theorem because the corresponding sides are proportional. omplete the ratios to verify Side-Side-Side (SSS) Triangle Similarity. a : b : c = ka : kb : kc You can use the Pythagorean Theorem to compare the lengths of the sides of JKL. What must be true of the set of numbers ka, kb, and kc? (ka ) + (kb ) = k a + k b = k (a + b ) = k ( c ) = (kc) The set of numbers ka, kb, and kc form a Pythagorean triple. Houghton Mifflin Harcourt Publishing ompany Module 74 Lesson
7 Reflect 0. Suppose you are given a right triangle with two side lengths. What would have to be true for you to use a Pythagorean triple to find the remaining side length? The given side lengths would have to be two numbers in a Pythagorean triple. lso, if the legs are given, the side lengths would have to be the smaller two numbers in the triple, whereas if one leg and the otenuse are given, the side lengths would have to include the largest number in the triple. Your Turn Use Pythagorean triples to find the unknown side length.. R. In XYZ, the otenuse XY has length 68, and the shorter leg XZ has length. P Q 8 8 : 4 = : 4, so the side lengths are multiples of the Pythagorean triple, 4, 5. : 4 : 5 = 6 () : 6 (4) : 6 (5) = 8 : 4 : 0, so PR = 0. Elaborate 4 : 68 = 8 : 7, so the side lengths are multiples of 8, 5, and 7. 8 : 5 : 7 = 4 (8) : 4 (5) : 4 (7) = : 60 : 68, so YZ = 60.. escribe the type of problems involving special right triangles you can solve. Once you identify the side (longer leg, shorter leg, otenuse) that is given, you can find the lengths of the other two sides by applying relationships such as the length of the otenuse in a triangle being twice as long as the length of the shorter leg. Houghton Mifflin Harcourt Publishing ompany 4. How can you use Pythagorean triples to solve right triangles? Suppose two side lengths of a right triangle are given and correspond to two numbers in a Pythagorean triple. If one of the given sides is the otenuse and one of the lengths is the largest number of the triple, then the unknown length is the remaining number in the triple. Likewise, if neither given side is the otenuse and the lengths are the smaller two numbers of the triple, the unknown side length must be the largest number of the triple. 5. iscussion How many Pythagorean triples are there? Infinitely many; for eample, since, 4, and 5 is a Pythagorean triple, so are () = 6, (4) = 8, and (5) = 0; 9,, and 5;, 6, and 0; and so on. 6. Essential Question heck-in What is the ratio of the length of the otenuse to the length of the shorter leg in any triangle? The ratio is to. Module 75 Lesson
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