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

Do Now 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 TSW justify and apply properties of 45°-45°-90° 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°. 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.

Example 1: Finding Side Lengths in a 45°- 45º- 90º Triangle Find the value of x. Give your answer in simplest radical form.

Example 2: Finding Side Lengths in a 45º- 45º- 90º Triangle Find the value of x. Give your answer in simplest radical form.

Example 3 Find the value of x. Give your answer in simplest radical form.

Example 4 Find the value of x. Give your answer in simplest radical form.

Example 5: Craft Application Eric is cutting a square of material for a tablecloth. The table’s diagonal is 36 inches. He 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 Eric cut to make the tablecloth? Round to the nearest inch.

Example 6: Application Sarah wants to make a bandana for her dog by folding a square cloth into a 45°-45°-90° triangle. Her dog’s neck has a circumference of about 32 cm. The folded bandana needs to be an extra 16 cm long so Sarah can tie it around her dog’s neck. What should the side length of the square be? Round to the nearest centimeter.  

Example 7 Caelyn’s dog is wearing a square bandana with a side length of 42 cm. What would you expect the circumference of her dog’s neck to be? Round to the nearest centimeter.

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.

Example 8: Finding Side Lengths in a 30º-60º-90º Triangle Find the values of x and y. Give your answers in simplest radical form.

Example 9: Finding Side Lengths in a 30º-60º-90º Triangle Find the values of x and y. Give your answers in simplest radical form.

Example 10 Find the values of x and y. Give your answers in simplest radical form.

Example 11 Find the values of x and y. Give your answers in simplest radical form.

Example 12 Find the values of x and y. Give your answers in simplest radical form.

Example 13 Find the values of x and y. Give your answers in simplest radical form.

Example 14: Using the 30º-60º-90º Triangle Theorem An 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?

Example 15: Application The frame of the clock shown is an equilateral triangle. The length of one side of the frame is 20 cm. Will the clock fit on a shelf that is 18 cm below the shelf above it?

Example 16: Application 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.

Lesson Quiz: Part I Find the values of the variables. Give your answers in simplest radical form. 1. 2. 3. 4. x = 10; y = 20

Lesson Quiz: Part II Find the perimeter and area of each figure. Give your answers in simplest radical form. 5. a square with diagonal length 20 cm 6. an equilateral triangle with height 24 in.