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Published byAbigail Atkinson Modified over 6 years ago

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

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**Objectives Justify and apply properties of 45°-45°-90° triangles.**

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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. 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.

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In a triangle, if you are given the length of a leg, the other leg is the same length as it is an isosceles right triangle. If you are given the length of a leg, the length of the hypotenuse is the length of the leg times the square root of 2. If you are given the length of the hypotenuse, divide this length by the square root of two, simplify, and this will give you the length of the leg.

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**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.

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**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.

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Check It Out! Example 1a 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 x = 20 Simplify.

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Check It Out! Example 1b 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 16. Rationalize the denominator.

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**A 30°-60°-90° triangle is another special right triangle**

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

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In a triangle, if you are given the length of the short leg, the length of the hypotenuse is twice the length of the short leg. The length of the long leg is the length of the short leg times the square root of three. If you are given the length of the hypotenuse, the length of the short leg is the length of the hypotenuse divided by 2. Then multiply the length of the short leg by the square root of three to get the long leg. If you are given the long leg, divide the length of the long leg by the square root of three, simplify and you will have the length of the short leg. Then multiply the length of the short leg by 2 to get the hypotenuse. Remember, by the Triangle InEquality Theorem the longest side, the hypotenuse, is opposite the biggest, 90 degree angle. The shortest side is opposite the smallest angle, the 30 degree angle.

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**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.

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**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.

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Check It Out! Example 3a Find the values of x and y. Give your answers in simplest radical form. Hypotenuse = 2(shorter leg) Divide both sides by 2. y = 27 Substitute for x.

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Check It Out! Example 3b Find the values of x and y. Give your answers in simplest radical form. y = 2(5) y = 10 Simplify.

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Check It Out! Example 3c 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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Check It Out! Example 3d Find the values of x and y. Give your answers in simplest radical form. Rationalize the denominator. x = 2y Hypotenuse = 2(shorter leg) Simplify.

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Lesson Quiz: Part I Find the values of the variables. Give your answers in simplest radical form. x = 10; y = 20

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