# 9-4 Applying Special Right Triangles Holt Geometry Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz Lesson Quiz

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• 9-4Applying Special Right TrianglesHolt GeometryWarm UpLesson PresentationLesson Quiz

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

1. 2.

Simplify each expression.

3.4.9.4 Special Right Triangles

• Justify and apply properties of 45-45-90 triangles.

Justify and apply properties of 30- 60- 90 triangles.Objectives9.4 Special Right Triangles

• 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.9.4 Special Right Triangles

• 9.4 Special Right Triangles

• Example 1A: Finding Side Lengths in a 45- 45- 90 TriangleFind 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 8.9.4 Special Right Triangles

• Example 1B: Finding Side Lengths in a 45- 45- 90 TriangleFind 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.9.4 Special Right Triangles

• Check It Out! Example 1a Find the value of x. Give your answer in simplest radical form.x = 20Simplify.9.4 Special Right Triangles

• 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.9.4 Special Right Triangles

• Example 2: Craft ApplicationJana is cutting a square of material for a tablecloth. The tables diagonal is 36 inches. She wants the diagonal of the tablecloth to be an extra 10 inches so it will hang over the edges of the table. What size square should Jana cut to make the tablecloth? Round to the nearest inch. Jana needs a 45-45-90 triangle with a hypotenuse of 36 + 10 = 46 inches.9.4 Special Right Triangles

• Check It Out! Example 2 What if...? Tessas other dog is wearing a square bandana with a side length of 42 cm. What would you expect the circumference of the other dogs neck to be? Round to the nearest centimeter.Tessa needs a 45-45-90 triangle with a hypotenuse of 42 cm.9.4 Special Right Triangles

• A 30-60-90 triangle is another special right triangle. You can use an equilateral triangle to find a relationship between its side lengths.9.4 Special Right Triangles

• Example 3A: Finding Side Lengths in a 30-60-90 TriangleFind the values of x and y. Give your answers in simplest radical form.Hypotenuse = 2(shorter leg)22 = 2xDivide both sides by 2.11 = xSubstitute 11 for x.9.4 Special Right Triangles

• Example 3B: Finding Side Lengths in a 30-60-90 TriangleFind the values of x and y. Give your answers in simplest radical form.Rationalize the denominator.Hypotenuse = 2(shorter leg).Simplify.y = 2x9.4 Special Right Triangles

• 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 = 279.4 Special Right Triangles

• Check It Out! Example 3b Find the values of x and y. Give your answers in simplest radical form.Simplify.y = 2(5)y = 109.4 Special Right Triangles

• Check It Out! Example 3c Find the values of x and y. Give your answers in simplest radical form.Hypotenuse = 2(shorter leg)Divide both sides by 2.Substitute 12 for x.24 = 2x12 = x9.4 Special Right Triangles

• Check It Out! Example 3d Find the values of x and y. Give your answers in simplest radical form.Rationalize the denominator.Hypotenuse = 2(shorter leg)x = 2ySimplify.9.4 Special Right Triangles

• Example 4: Using the 30-60-90 Triangle TheoremAn ornamental pin is in the shape of an equilateral triangle. The length of each side is 6 centimeters. Josh will attach the fastener to the back along AB. Will the fastener fit if it is 4 centimeters long?Step 1 The equilateral triangle is divided into two 30-60-90 triangles.The height of the triangle is the length of the longer leg. 9.4 Special Right Triangles

• Example 4 ContinuedStep 2 Find the length x of the shorter leg.Step 3 Find the length h of the longer leg.The pin is approximately 5.2 centimeters high. So the fastener will fit.Hypotenuse = 2(shorter leg)6 = 2x3 = xDivide both sides by 2.9.4 Special Right Triangles

• Check It Out! Example 4 What if? A manufacturer wants to make a larger clock with a height of 30 centimeters. What is the length of each side of the frame? Round to the nearest tenth.Step 1 The equilateral triangle is divided into two 30-60-90 triangles.The height of the triangle is the length of the longer leg. 9.4 Special Right Triangles

• Check It Out! Example 4 ContinuedStep 2 Find the length x of the shorter leg.Each side is approximately 34.6 cm.Step 3 Find the length y of the longer leg.Rationalize the denominator.Hypotenuse = 2(shorter leg)y = 2xSimplify.9.4 Special Right Triangles

• Lesson Quiz: Part IFind the values of the variables. Give your answers in simplest radical form.1. 2.

3. 4.

x = 10; y = 209.4 Special Right Triangles

• Lesson Quiz: Part IIFind the perimeter and area of each figure. Give your answers in simplest radical form.

5. a square with diagonal length 20 cm6. an equilateral triangle with height 24 in.

9.4 Special Right Triangles

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