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Five-Minute Check (over Lesson 8–2) Then/Now Theorem 8.8: 45°-45°-90° Triangle Theorem Example 1: Find the Hypotenuse Length in a 45°-45°-90° Triangle Example 2: Find the Leg Lengths in a 45°-45°-90° Triangle Theorem 8.9: 30°-60°-90° Triangle Theorem Example 3: Find Lengths in a 30°-60°-90° Triangle Example 4: Real-World Example: Use Properties of Special Right Triangles Lesson Menu

Find x. A. 5 B. C. D. 10.5 A B C D 5-Minute Check 1

Find x. A. B. C. 45 D. 51 A B C D 5-Minute Check 2

Determine whether ΔQRS with vertices Q(2, –3), R(0, –1), and S(4, –1) is a right triangle. If so, identify the right angle. A. yes; S B. yes; Q C. yes; R D. no A B C D 5-Minute Check 3

Determine whether the set of numbers can be the measures of the sides of a triangle. If so, classify the triangle as acute, obtuse, or right. 16, 30, 33 A. yes, acute B. yes, obtuse C. yes, right D. no A B C D 5-Minute Check 4

Determine whether the set of numbers can be the measures of the sides of a triangle. If so, classify the triangle as acute, obtuse, or right. A. yes, acute B. yes, obtuse C. yes, right D. no A B C D 5-Minute Check 5

A B C D Which of the following are the lengths of an acute triangle? __ 1 2 A B C D 5-Minute Check 6

Use the properties of 45°-45°-90° triangles. You used properties of isosceles and equilateral triangles. (Lesson 4–6) Use the properties of 45°-45°-90° triangles. Use the properties of 30°-60°-90° triangles. Then/Now

Concept

Find the Hypotenuse Length in a 45°-45°-90° Triangle A. Find x. The given angles of this triangle are 45° and 90°. This makes the third angle 45°, since 180 – 45 – 90 = 45. Thus, the triangle is a 45°-45°-90° triangle. Example 1

45°-45°-90° Triangle Theorem Find the Hypotenuse Length in a 45°-45°-90° Triangle 45°-45°-90° Triangle Theorem Substitution Example 1

Find the Hypotenuse Length in a 45°-45°-90° Triangle B. Find x. The legs of this right triangle have the same measure, x, so it is a 45°-45°-90° triangle. Use the 45°-45°-90° Triangle Theorem. Example 1

45°-45°-90° Triangle Theorem Find the Hypotenuse Length in a 45°-45°-90° Triangle 45°-45°-90° Triangle Theorem Substitution x = 12 Answer: x = 12 Example 1

A. Find x. A. 3.5 B. 7 C. D. A B C D Example 1

B. Find x. A. B. C. 16 D. 32 A B C D Example 1

45°-45°-90° Triangle Theorem Find the Leg Lengths in a 45°-45°-90° Triangle Find a. The length of the hypotenuse of a 45°-45°-90° triangle is times as long as a leg of the triangle. 45°-45°-90° Triangle Theorem Substitution Example 2

Rationalize the denominator. Find the Leg Lengths in a 45°-45°-90° Triangle Divide each side by Rationalize the denominator. Multiply. Divide. Example 2

Find b. A. B. 3 C. D. A B C D Example 2

Concept

Find Lengths in a 30°-60°-90° Triangle Find x and y. The acute angles of a right triangle are complementary, so the measure of the third angle is 90 – 30 or 60. This is a 30°-60°-90° triangle. Example 3

Find the length of the longer side. Find Lengths in a 30°-60°-90° Triangle Find the length of the longer side. 30°-60°-90° Triangle Theorem Substitution Simplify. Example 3

Find the length of hypotenuse. Find Lengths in a 30°-60°-90° Triangle Find the length of hypotenuse. 30°-60°-90° Triangle Theorem Substitution Simplify. Answer: x = 4, Example 3

Find BC. A. 4 in. B. 8 in. C. D. 12 in. A B C D Example 3

Use Properties of Special Right Triangles QUILTING A quilt has the design shown in the figure, in which a square is divided into 8 isosceles right triangles. If the length of one side of the square is 3 inches, what are the dimensions of each triangle? Example 4

Use Properties of Special Right Triangles Understand You know that the length of the side of the square equals 3 inches. You need to find the length of the side and hypotenuse of one isosceles right triangle. Plan Find the length of one side of the isosceles right triangle, and use the 45°-45°-90° Triangle Theorem to find the hypotenuse. Example 4

So the side length is 1.5 inches. Use Properties of Special Right Triangles Solve Divide the length of the side of the square by 2 to find the length of the side of one triangle. 3 ÷ 2 = 1.5 So the side length is 1.5 inches. 45°-45°-90° Triangle Theorem Substitution Example 4

Answer: The side length is 1.5 inches and the hypotenuse is Use Properties of Special Right Triangles Answer: The side length is 1.5 inches and the hypotenuse is Check Use the Pythagorean Theorem to check the dimensions of the triangle. ? 2.25 + 2.25 = 4.5 ? 4.5 = 4.5  Example 4

BOOKENDS Shaina designed 2 identical bookends according to the diagram below. Use special triangles to find the height of the bookends. A. B. 10 C. 5 D. A B C D Example 4

End of the Lesson