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

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

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

4. Isosceles right triangle are commonly called

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

7. Find the value of x. Give your answer in simplest radical form.
Rationalize the denominator.

8. Find the value of x.
Give your answer in
simplest radical form.
Simplify.
x = 20

9. Find the value of x. Give your answer in simplest radical form.
Rationalize the denominator.

10. Jana is cutting a square of material for a tablecloth
Jana is cutting a square of material for a tablecloth. The table’s 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 = 46 inches.

11. What if...? Tessa’s other dog is wearing a square bandana with a side length of 42 cm. What would you expect the circumference of the other dog’s neck to be? Round to the nearest centimeter.
Tessa needs a 45°-45°-90° triangle with a hypotenuse of 42 cm.

12. A 30°-60°-90° triangle is another special right triangle.
Half of an equilateral triangle is a 30°-60°-90°
Long Leg (LL)
Short Leg (SL)
Hypotenuse (H)

13. Hypotenuse = (shorter leg)2

14. Find the values of x and y. Give your answers in simplest radical form.
Short leg
22 = 2x
Hypotenuse = 2(shorter leg)
11 = x
Divide both sides by 2.
Long leg
Substitute 11 for x.

15. Find the values of x and y. Give your answers in simplest radical form.
Short leg
Rationalize the denominator.
Hypotenuse
Hypotenuse = 2(shorter leg).
y = 2x
Simplify.

16. 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.

17. Find the values of x and y. Give your answers in simplest radical form.
Simplify.

18. Find the values of x and y. Give your answers in simplest radical form.
Hypotenuse = 2(shorter leg)
12 = x
Divide both sides by 2.
Substitute 12 for x.

19. Find the values of x and y. Give your answers in simplest radical form.
Rationalize the denominator.
x = 2y
Hypotenuse = 2(shorter leg)
Simplify.

20. Example 4: 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?
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.

21. Example 4 Continued
Step 2 Find the length x of the shorter leg.
6 = 2x
Hypotenuse = 2(shorter leg)
3 = x
Divide both sides by 2.
Step 3 Find the length h of the longer leg.
The pin is approximately 5.2 centimeters high. So the fastener will fit.

22. 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 30º-60º-90º triangles.
The height of the triangle is the length of the longer leg.

23. Check It Out! Example 4 Continued
Step 2 Find the length x of the shorter leg.
Rationalize the denominator.
Step 3 Find the length y of the longer leg.
y = 2x
Hypotenuse = 2(shorter leg)
Simplify.
Each side is approximately 34.6 cm.

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

25. 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.