Special Right Triangles

Transcription

1 . Special Right Triangles Essential Question What is the relationship among the side lengths of triangles? triangles? Side Ratios of an Isosceles Right Triangle ATTENDING TO PRECISION To be proficient in math, ou need to epress numerical answers with a degree of precision appropriate for the problem contet. Work with a partner. a. Use dnamic geometr software to construct an isosceles right triangle with a leg length of units. b. Find the acute angle measures. Eplain wh this triangle is called a triangle. c. Find the eact ratios of the side lengths (using square roots). AC = AC 0 C A 0 d. Repeat parts (a) and (c) for several other isosceles right triangles. Use our results to write a conjecture about the ratios of the side lengths of an isosceles right triangle. B Sample Points A(0, ) B(, 0) C(0, 0) Segments =.66 AC = Angles m A = m B = Side Ratios of a Triangle Work with a partner. a. Use dnamic geometr software to construct a right triangle with acute angle measures of and (a triangle), where the shorter leg length is units. b. Find the eact ratios Sample A of the side lengths (using square roots). AC = AC 0 C 0 c. Repeat parts (a) and (b) for several other triangles. Use our results to write a conjecture about the ratios of the side lengths of a triangle. Communicate Your Answer. What is the relationship among the side lengths of triangles? triangles? B Points A(0,.0) B(, 0) C(0, 0) Segments = 6 AC =.0 Angles m A = m B = Section. Special Right Triangles

2 . Lesson What You Will Learn Core Vocabular Previous isosceles triangle Find side lengths in special right triangles. Solve real-life problems involving special right triangles. Finding Side Lengths in Special Right Triangles A triangle is an isosceles right triangle that can be formed b cutting a square in half diagonall. REMEMBER An epression involving a radical with inde is in simplest form when no radicands have perfect squares as factors other than, no radicands contain fractions, and no radicals appear in the denominator of a fraction. Theorem Theorem Triangle Theorem In a triangle, the hpotenuse is times as long as each leg. Finding Side Lengths in Triangles Find the value of. Write our answer in simplest form. a. b. 8 Proof E., p. 6 hpotenuse = leg a. B the Triangle Sum Theorem (Theorem.), the measure of the third angle must be, so the triangle is a triangle. hpotenuse = leg Triangle Theorem = 8 = 8 The value of is 8. Substitute. Simplif. b. B the Base Angles Theorem (Theorem.6) and the Corollar to the Triangle Sum Theorem (Corollar.), the triangle is a triangle. hpotenuse = leg Triangle Theorem = Substitute. = Divide each side b. = Simplif. The value of is. Chapter Right Triangles and Trigonometr

3 Theorem Theorem Triangle Theorem In a triangle, the hpotenuse is twice as long as the shorter leg, and the longer leg is times as long as the shorter leg. Proof E., p. 6 hpotenuse = shorter leg longer leg = shorter leg REMEMBER Because the angle opposite is larger than the angle opposite, the leg with length is longer than the leg with length b the Triangle Larger Angle Theorem (Theorem 6.0). Finding Side Lengths in a Triangle Find the values of and. Write our answer in simplest form. Step Find the value of. longer leg = shorter leg = Substitute Triangle Theorem = Divide each side b. = Multipl b. = Multipl fractions. = Simplif. The value of is. Step Find the value of. hpotenuse = shorter leg = = Triangle Theorem Substitute. Simplif. The value of is 6. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Find the value of the variable. Write our answer in simplest form..... h Section. Special Right Triangles

4 Solving Real-Life Problems Modeling with Mathematics The road sign is shaped like an equilateral triangle. Estimate the area of the sign b finding the area of the equilateral triangle. First find the height h of the triangle b dividing it into two triangles. The length of the longer leg of one of these triangles is h. The length of the shorter leg is 8 inches. h = 8 = Triangle Theorem Use h = 8 to find the area of the equilateral triangle. Area = bh = (6) ( 8 ) 6.8 The area of the sign is about 6 square inches. 6 in. YIELD 8 in. 8 in. h 6 in. 6 in. Finding the Height of a Ramp A tipping platform is a ramp used to unload trucks. How high is the end of an 80-foot ramp when the tipping angle is?? ramp 80 ft height of ramp tipping angle ft When the tipping angle is, the height h of the ramp is the length of the shorter leg of a triangle. The length of the hpotenuse is 80 feet. 80 = h Triangle Theorem 0 = h Divide each side b. When the tipping angle is, the height h of the ramp is the length of a leg of a triangle. The length of the hpotenuse is 80 feet. 80 = h Triangle Theorem 80 = h Divide each side b. 6.6 h Use a calculator. When the tipping angle is, the ramp height is 0 feet. When the tipping angle is, the ramp height is about 6 feet inches. Monitoring Progress Help in English and Spanish at BigIdeasMath.com. The logo on a reccling bin resembles an equilateral triangle with side lengths of 6 centimeters. Approimate the area of the logo. 6. The bod of a dump truck is raised to empt a load of sand. How high is the -foot-long bod from the frame when it is tipped upward b a angle? Chapter Right Triangles and Trigonometr

5 . Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check. VOCULARY Name two special right triangles b their angle measures.. WRITING Eplain wh the acute angles in an isosceles right triangle alwas measure. Monitoring Progress and Modeling with Mathematics In Eercises 6, find the value of. Write our answer in simplest form. (See Eample.) In Eercises 0, find the values of and. Write our answers in simplest form. (See Eample.) ERROR ANALYSIS In Eercises and, describe and correct the error in finding the length of the hpotenuse.. B the Triangle Sum Theorem (Theorem.), the measure of the third angle must be. So, the triangle is a triangle. hpotenuse = shorter leg =. B the Triangle Sum Theorem (Theorem.), the measure of the third angle must be. So, the triangle is a triangle. hpotenuse = leg leg = So, the length of the hpotenuse is units. In Eercises and, sketch the figure that is described. Find the indicated length. Round decimal answers to the nearest tenth.. The side length of an equilateral triangle is centimeters. Find the length of an altitude.. The perimeter of a square is 6 inches. Find the length of a diagonal. In Eercises and 6, find the area of the figure. Round decimal answers to the nearest tenth. (See Eample.). 8 ft 6. m m. PROBLEM SOLVING Each half of the drawbridge is about 8 feet long. How high does the drawbridge rise when is??? (See Eample.) 8 ft m m So, the length of the hpotenuse is units. Section. Special Right Triangles

6 8. MODELING WITH MATHEMATICS A nut is shaped like a regular heagon with side lengths of centimeter. Find the value of. (Hint: A regular heagon can be divided into si congruent triangles.) cm. THOUGHT PROVOKING A special right triangle is a right triangle that has rational angle measures and each side length contains at most one square root. There are onl three special right triangles. The diagram below is called the Ailles rectangle. Label the sides and angles in the diagram. Describe all three special right triangles.. PROVING A THEOREM Write a paragraph proof of the Triangle Theorem (Theorem.). Given DEF is a D triangle. Prove The hpotenuse is times as long as each leg. F E 0. HOW DO YOU SEE IT? The diagram shows part of the Wheel of Theodorus. 6. WRITING Describe two was to show that all isosceles right triangles are similar to each other.. MAKING AN ARGUMENT Each triangle in the diagram is a triangle. At Stage 0, the legs of the triangle are each unit long. Your brother claims the lengths of the legs of the triangles added are halved at each stage. So, the length of a leg of a triangle added in Stage 8 will be unit. Is our 6 brother correct? Eplain our reasoning. a. Which triangles, if an, are triangles? b. Which triangles, if an, are triangles? Stage 0 Stage Stage. PROVING A THEOREM Write a paragraph proof of the Triangle Theorem (Theorem.). (Hint: Construct JML congruent to JKL.) Given JKL is a triangle. Prove The hpotenuse is twice as long as the shorter leg, and the longer leg is times as long as the shorter leg. J K L M Stage Stage. USING STRUCTURE TUV is a triangle, where two vertices are U(, ) and V(, ), UV is the hpotenuse, and point T is in Quadrant I. Find the coordinates of T. Maintaining Mathematical Proficienc Find the value of. (Section 8.) 6. DEF LMN. C QRS Reviewing what ou learned in previous grades and lessons E D N 0 F 0 L M A. B S C R Q 6 Chapter Right Triangles and Trigonometr