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5-8

5-8. Applying Special Right Triangles. Warm Up. Lesson Presentation. Lesson Quiz. Holt Geometry. Warm Up For Exercises 1 and 2, find the value of x . Give your answer in simplest radical form. 1. 2. Simplify each expression. 3. 4. Objectives.

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5-8

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  1. 5-8 Applying Special Right Triangles Warm Up Lesson Presentation Lesson Quiz Holt Geometry

  2. Warm Up For Exercises 1 and 2, find the value of x. Give your answer in simplest radical form. 1.2. Simplify each expression. 3.4.

  3. Objectives Justify and apply properties of 45°-45°-90° triangles. Justify and apply properties of 30°- 60°- 90° triangles.

  4. A diagonal of a square divides it into two congruent isosceles right triangles. Since the base angles of an isosceles triangle are congruent, the measure of each acute angle is 45°. So another name for an isosceles right triangle is a 45°-45°-90° triangle. 45 45 A 45°-45°-90° triangle is one type of special right triangle. You can use the Pythagorean Theorem to find a relationship among the side lengths of a 45°-45°-90° triangle.

  5. l = h The Isosceles Right 50 Why? a2 + b2 = c2 l2 + l2 = h2 2l2 = h2

  6. Corollary to Theorem 50: Each diagonal of a square is times the length of one side. 45 45

  7. Example 1A: Finding Side Lengths in a 45°- 45º- 90º Triangle Find the value of x. Give your answer in simplest radical form. By the Triangle Sum Theorem, the measure of the third angle in the triangle is 45°. So it is a 45°-45°-90° triangle with a leg length of 8.

  8. Example 1B: Finding Side Lengths in a 45º- 45º- 90º Triangle Find the value of x. Give your answer in simplest radical form. The triangle is an isosceles right triangle, which is a 45°-45°-90° triangle. The length of the hypotenuse is 5. Rationalize the denominator.

  9. By the Triangle Sum Theorem, the measure of each of the two congruent angles in the triangle is 45°. So it is a 45°-45°-90° triangle with a leg length of Check It Out! Example 1a Find the value of x. Give your answer in simplest radical form. Simplify. x = 20

  10. A 30°-60°-90° triangle is another special right triangle. You can use an equilateral triangle to find a relationship between its side lengths. 30 60 60

  11. Right ^ 51 This can also be proven algebraically, but let’s move on to some problems. In your reading, notice the corollary to Theorem 51. This corollary involves equilateral triangles and a familiar formula.

  12. Example 3A: Finding Side Lengths in a 30º-60º-90º Triangle Find the values of x and y. Give your answers in simplest radical form. 22 = 2x Hypotenuse = 2(shorter leg) 11 = x Divide both sides by 2. Substitute 11 for x.

  13. Example 3B: Finding Side Lengths in a 30º-60º-90º Triangle Find the values of x and y. Give your answers in simplest radical form. Rationalize the denominator. y = 2x Hypotenuse = 2(shorter leg). Simplify.

  14. Example 3C: Finding Side Lengths in a 30º-60º-90º Triangle Find the values of x and y. Give your answers in simplest radical form. 24 = 2x Hypotenuse = 2(shorter leg) 12 = x Divide both sides by 2. Substitute 12 for x.

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