Name Class Date. Investigating an Isosceles Right Triangle. x B Let the legs of the right triangle have length x. You can use the Pythagorean

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1 Name lass ate pplying Special Right Triangles Going eeper Essential question: What can you say about the side lengths associated with special right triangles? 5-8 There are two special right triangles that arise frequently in problem-solving situations. It is useful to know the relationships among the side lengths of these triangles. G-SRT.3.6 EXPLORE Investigating an Isosceles Right Triangle The figure shows an isosceles right triangle. What is the measure of each base angle of the triangle? Why? Let the legs of the right triangle have length. You can use the Pythagorean Theorem to find the length of the hypotenuse in terms of. 2 = Pythagorean Theorem 2 = ombine like terms. = Find the square root of both sides and simplify. Houghton Mifflin Harcourt Publishing ompany REFLET a. student claims that if you know one side length of an isosceles right triangle, then you know all the side lengths. o you agree or disagree? Eplain. b. Eplain how to find y in the right triangle at right. y hapter Lesson 8

2 2 G-SRT.3.6 EXPLORE Investigating nother Special Right Triangle In the figure, is an equilateral triangle and is a perpendicular from to. Eplain how to find the angle measures in. Eplain why. Let the length of be. What is the length of? Why? In the space below, show how to use the Pythagorean Theorem to find the length of. REFLET 2a. What is the ratio of the side lengths in a right triangle with acute angles that measure 30 and 60? 2b. Error nalysis student drew a right triangle with a 60 angle and a hypotenuse of length 0. Then he labeled the other side lengths as shown. Eplain how you can tell just by glancing at the side lengths that the student made an error. Then eplain the error. J 0 L 60 5 K 0 3 Houghton Mifflin Harcourt Publishing ompany hapter Lesson 8

3 The right triangles you investigated are sometimes called and right triangles. The side-length relationships that you discovered can be used to find lengths in any such triangles. 3 G-SRT.3.8 EXMPLE Solving Special Right Triangles Refer to the diagram of the triangle. Fill in the calculations that help you find the missing side lengths. Give answers in simplest radical form. 45 = = = = = 45 7 Refer to the diagram of the triangle. Fill in the calculations that help you find the missing side lengths. Give answers in simplest radical form. E = E F = E = EF = F = dd the side lengths you calculated to the diagrams. 60 F Houghton Mifflin Harcourt Publishing ompany REFLET 3a. Suppose you are given the length of the hypotenuse of a triangle. How can you calculate the length of a leg? 3b. Suppose you are given the length of the longer leg of a triangle. How can you calculate the length of the shorter leg? 3c. When finding a leg length in a triangle, one student gave the answer 30 2 and another gave the answer 5 2. Show the answers are equivalent. hapter Lesson 8

4 PRTIE Find the value of. Give your answer in simplest radical form.. S 2. J R 45 3 T K 4 L 4. E 5. M N W F L U 30 V 7. S 8. E 5 F 9. 7 R 60 T 0. Error nalysis Two students were asked to find the value of in the figure at right. Which student s work is correct? Eplain the other student s error. Roberto s Work In a triangle, the hypotenuse is twice as long as the shorter leg, so = 4. The ratio of the lengths of the legs is : 3, so = 4 3. G aron s Work In a triangle, the side lengths are in a ratio of : 3 :2, so must be 3 times the length of. Therefore, = M 30 K L 60 8 Houghton Mifflin Harcourt Publishing ompany hapter Lesson 8

5 Name lass ate dditional Practice 5-8 Find the value of in each figure. Give your answer in simplest radical form Find the values of and y. Give your answers in simplest radical form. 4. = y = 5. = y = 6. = y = Lucia is an archaeologist trekking through the jungle of the Yucatan Peninsula. She stumbles upon a stone structure covered with creeper vines and ferns. She immediately begins taking measurements of her discovery. (Hint: rawing some figures may help.) 7. round the perimeter of the building, Lucia finds small alcoves at regular intervals carved into the stone. The alcoves are triangular in shape with a horizontal base and two sloped equal-length sides that meet at a right angle. Each of the sloped sides measures 4 4 Houghton Mifflin Harcourt Publishing ompany inches. Lucia has also found several stone tablets inscribed with characters. The stone tablets measure 22 inches long. Lucia hypothesizes that the alcoves once held the stone 8 tablets. Tell whether Lucia s hypothesis may be correct. Eplain your answer Lucia also finds several statues around the building. The statues measure 9 inches 6 tall. She wonders whether the statues might have been placed in the alcoves. Tell whether this is possible. Eplain your answer. hapter 5 23 Lesson 8

6 Problem Solving. In bowling, the pins are arranged in a pattern based on equilateral triangles. What is the distance between pins and 5? 2. To secure an outdoor canopy, a 64-inch cord is etended from the top of a vertical pole to the ground. If the cord makes a 60 angle with the ground, how tall is the pole? n equilateral triangle has an altitude of 6. shelf is an isosceles right triangle, and 2 inches. What is the side length of the longest side is 38 centimeters. What the triangle? is the length of each of the other two sides? ssume is in the first quadrant, with m = Suppose that is a leg of, a triangle. What are possible coordinates of point? (6, 4.5) (6, 2) (7, 2) (8, 7) 8. Suppose is a triangle and is the side opposite the 60 angle. What are the approimate coordinates of point? F (4.9, 2) H (8.7, 2) G (4.5, 2) J (7., 2) Houghton Mifflin Harcourt Publishing ompany hapter Lesson 8