Special Right Triangles LESSON 8–3

Lesson Menu Five-Minute Check (over Lesson 8–2) TEKS 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

Over Lesson 8–2 5-Minute Check 1 Find x. A.5 B. C. D.10.5

Over Lesson 8–2 5-Minute Check 2 Find x. A. B. C.45 D.51

Over Lesson 8–2 5-Minute Check 4 A.yes, acute B.yes, obtuse C.yes, right D.no 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

Over Lesson 8–2 5-Minute Check 5 A.yes, acute B.yes, obtuse C.yes, right D.no 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.

Over Lesson 8–2 5-Minute Check 6 Which of the following are the lengths of an acute triangle? A.25, 20, 15 B.4, 7, 8 C.0.7, 2.4, 2.5 D.36, 48, 62 __ 1 2

TEKS Targeted TEKS G.9(B) Apply the relationships in special right triangles 30º-60º- 90º and 45º-45º-90º and the Pythagorean theorem, including Pythagorean triples, to solve problems. Mathematical Processes G.1(E), G.1(F)

Then/Now You used properties of isosceles and equilateral triangles. Use the properties of 45°-45°-90° triangles. Use the properties of 30°-60°-90° triangles.

Concept

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

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

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

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

Example 2 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. Substitution 45°-45°-90° Triangle Theorem

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

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

Concept

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

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

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

Example 4 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 AnalyzeYou 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. FormulateFind the length of one side of the isosceles right triangle, and use the 45°-45°-90° Triangle Theorem to find the hypotenuse.

Example 4 Use Properties of Special Right Triangles DetermineDivide 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 Use Properties of Special Right Triangles JustifyUse the Pythagorean Theorem to check the dimensions of the triangle. ? = 4.5 ? 4.5 = 4.5 EvaluateWe used the properties of special triangles. Our answer seems reasonable. Answer:The side length is 1.5 inches and the hypotenuse is

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.

Special Right Triangles LESSON 8–3