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Special Right Triangles 8-2 1 8-2 Special Right Triangles 8-2 1. Plan What You’ll Learn Check Skills You’ll Need GO for Help Lesson 1-6 Objectives 1 To use the properties of • To use the properties of 458-458-908 triangles Use a protractor to find the measures of the angles of each triangle. 45°-45°-90° Triangles 2 To use the properties of • To use the properties of 1. 2. 3. 30°-60°-90° Triangles 308-608-908 triangles ͙ළ2 ͙ළ2 4 2 3͙ළ2 3 . And Why Examples 1 Finding the Length of the To find the distance from 2 ͙ළ 2 3 Hypotenuse home plate to second base 45, 45, 90 30, 60, 90 3 on a softball diamond, as 45, 45, 90 2 Finding the Length of a Leg in Example 3 3 Real-World Connection 4 Using the Length of One Side 5 Real-World Connection 1 Using 458-458-908 Triangles 1 Math Background 8 The acute angles of an isosceles right triangle are both 45 angles. Another name The ratio of the lengths of any 8 8 8 for an isosceles right triangle is a 45 -45 -90 triangle. If each leg has length x and two sides of a right triangle is a the hypotenuse has length y, you can solve for y in terms of x. function of either acute angle. x2 + x2 = y2 Use the Pythagorean Theorem. This can be proved using similarity x x theorems and is the basis for 2x2 = y2 Simplify. 45Њ 45Њ the six trigonometric functions. x 2 = y Take the square root of each side. y ! The fixed side-length ratios of You have just proved the following theorem. 45°-45°-90° and 30°-60°-90° triangles, easily found by applying the Pythagorean Theorem, Key Concepts Theorem 8-5 458-458-908 Triangle Theorem provide benchmark values for the trigonometric functions sine, In a 458-458-908 triangle, both legs are congruent and the s͙ළ2 45Њ cosine, and tangent of 30°, 45°, length of the hypotenuse is 2 times the length of a leg. s ! and 60° angles. Њ hypotenuse = 2 ? leg 45 D E ! s B C 1 A D E B C More Math Background: p. 414C 2 A D E B C 3 A D E B C 4 A D E B C 5 A D E B C Test-Taking Tip If you forget the 1 EXAMPLE Finding the Length of the Hypotenuse Lesson Planning and formula for a Resources 45°-45°-90° triangle, Find the value of each variable. use the Pythagorean See p. 414E for a list of the Theorem. The triangle a. b. 45Њ is isosceles, so the legs 9 resources that support this lesson. have the same length. ͙ළ Њ Њ x 2 2 45 45 45Њ h PowerPoint h = 2 ? 9 hypotenuse ≠ 2 ? leg x = 2 ? 2 2 Bell Ringer Practice ! ! ! ! h = 9 2 Simplify. x = 4 ! Check Skills You’ll Need For intervention, direct students to: Quick Check 1 Find the length of the hypotenuse of a 458-458-908 triangle with legs of length 53 . ! Measuring Angles 5 6 " Lesson 1-6: Example 2 Lesson 8-2 Special Right Triangles 425 Extra Skills, Word Problems, Proof Practice, Ch. 1 Special Needs L1 Below Level L2 For Example 3, have students check the answer by In the diagram for Theorem 8-6, construct a 30° angle cutting out a 60-mm by 60-mm square. They fold it adjacent to the 30° angle, using a leg as one side. along its diagonal, and measure the length of the Extend the base so that it intersects the new side. Discuss diagonal to the nearest millimeter. why this forms an equilateral triangle. learning style: tactile learning style: visual 425 2. Teach 2 EXAMPLE Finding the Length of a Leg Multiple Choice What is the value of x? 2 2 33 66 45Њ 6 Guided Instruction ! ! x 6 = 2 ? x hypotenuse ≠ 2 ? leg !6 ! x = Divide each side by 2. 45Њ 1 EXAMPLE Technology Tip 2 ! ! = 6 ? 2 = 6 2 Point out that using mental x 2 !2 !2 Multiply by a form of 1. ! ! math is much faster than using a x = 3 2 Simplify. calculator for part b. The calculator ! The correct answer is B. answer also would be inexact, whereas squaring the square root Quick Check 2 Find the length of a leg of a 458-458-908 triangle with a hypotenuse of length 10. of a number is always exact. 5 2 " 2 EXAMPLE Auditory Learners When you apply the 458-458-908 Triangle Theorem to a real-life example, you can Have several students explain use a calculator to evaluate square roots. aloud to the class how to rationalize a denominator. 3 EXAMPLE Real-World Connection PowerPoint Softball A high school softball diamond is a square. The distance from base to base is 60 ft. To the nearest foot, how far does a catcher throw the ball from home Additional Examples plate to second base? 1 Find the length of the The distance d from home plate to second 60 ft hypotenuse of a 45°-45°-90° base is the length of the hypotenuse of a triangle with legs of length 56 . 458-458-908 triangle. " Real-World Connection d 10 3 d = 60 2 hypotenuse ≠ 2 ? leg " Careers Opportunities for ! ! = 84.852814 2 Find the length of a leg coaching in women’s sports d Use a calculator. of a 45°-45°-90° triangle with have soared since the passage of Title IX in 1972. On a high school softball diamond, the catcher throws the ball about 85 ft from a hypotenuse of length 22. 11 2 " home plate to second base. 3 The distance from one corner Quick Check 3 A square garden has sides 100 ft long. You want to build a brick path along a to the opposite corner of a square diagonal of the square. How long will the path be? Round your answer to the playground is 96 ft. To the nearest nearest foot. 141 ft foot, how long is each side of the playground? 68 ft Visual Learners 12 Using 308-608-908 Triangles Suggest that students distinguish between the 45°-45°-90° and the Another type of special right triangle is a 308-608-908 triangle. 30°-60°-90° Triangle Theorems by using the “ratio” diagrams below. Key Concepts Theorem 8-6 308-608-908 Triangle Theorem In a 308-608-908 triangle, the length of the hypotenuse ° ͙3 30 2 is twice the length of the shorter leg. The length of the ͙2 30Њ 1 45° longer leg is 3 times the length of the shorter leg. 2s ! s͙ළ3 45° 60° hypotenuse = 2 ? shorter leg 1 1 60Њ longer leg = 3 ? shorter leg s Error Prevention! ! Whenever the length of a hypotenuse or longer leg of a 30°-60°-90° triangle is given, encourage students to find the 426 Chapter 8 Right Triangles and Trigonometry length of the shorter leg first. Advanced Learners L4 English Language Learners ELL After students learn and apply Theorem 8-5, have Ask students to complete each statement: The them write a formula for the area of an isosceles right shortest side of a triangle is always opposite the triangle whose hypotenuse has length s. smallest angle. In a 30°-60°-90° triangle, the shortest side is always opposite the 30° angle . 426 learning style: verbal learning style: verbal 4 EXAMPLE Math Tip To prove Theorem 8-6, draw a 308-608-908 triangle using an equilateral triangle. Students can use the Pythagorean Proof Proof of Theorem 8-6 W Theorem to check their work. For 308-608-908 #WXY in equilateral #WXZ, PowerPoint WY is the perpendicular bisector of XZ. Њ 1 1 30 Thus, XY = 2 XZ = 2 XW, or XW = 2XY = 2s. Additional Examples Also, 60Њ 60Њ 4 The longer leg of a 30°-60°-90° XY 2 + YW2 = XW2 Use the Pythagorean Theorem. X Z s Y triangle has length 18. Find the 2 + 2 = 2 s YW (2s) Substitute s for XY and 2s for XW. lengths of the shorter leg and the YW2 = 4s2 - s2 Subtract s2 from each side. hypotenuse. shorter leg: 63 ; " 2 = 2 hypotenuse: 12 3 YW 3s Simplify. " YW = s 3 Find the square root of each side. ! 5 A garden shaped like a rhombus has a perimeter of 100 ft and a 60° angle. Find the The 30°-60°-90° Triangle Theorem, like the 45°-45°-90° Triangle Theorem, lets you perpendicular height between two find two sides of a triangle when you know the length of the third side. bases. 21.7 ft 4 EXAMPLE Using the Length of One Side Resources • Daily Notetaking Guide 8-2 L3 Algebra Find the value of each variable. • Daily Notetaking Guide 8-2— d 5 5 = d 3 longer leg ≠ 3 ? shorter leg Adapted Instruction L1 ! Á 60Њ 30Њ 5 3 5 3 d = ? ! = ! Solve for d. f 3 3 3 " ! f = 2d hypotenuse ≠ 2 ? shorter leg Closure = ? 5 3 = 10 3 5 3 f 2 !3 3! Substitute !3 for d. In quadrilateral ABCD, AD = DC Њ Quick Check 4 Find the value of each variable. x = 4, y = 4 3 8 60 and AC = 20. Find the area of " x 30Њ ABCD. Leave your answer in y simplest radical form. C 5 EXAMPLE Real-World Connection 75° B Road Signs The moose warning sign at the left is an equilateral triangle. The height of the sign is 1 m.
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