1. 5-Minute Check on Lesson 7-2 Transparency 7-3 Click the mouse button or press the Space Bar to display the answers. Find x. 1. 2. 3. Determine whether ∆QRS with vertices Q(2,-3), R(0,-1), and S(4,-1) is a right triangle. If so, identify the right angle. Determine whether each set of measures forms a right triangle and state whether they form a Pythagorean triple. 4. 16, 30, 33 5. 6. Which of the following are not the lengths of sides of a right triangle? Standardized Test Practice: ACBD 25, 20, 154, 7.5, 8.50.7, 2.4, 2.5 36, 48, 62 12 7 x13 38 √95 ≈ 9.7 Yes,  Q No, No D √1613 ≈ 40.2 x Yes, No 5 3 13 ---, ---, ---- 8 2 8

2. Lesson 7-3 Special Case Right Triangles

3. Objectives Use properties of 45°- 45°- 90° triangles –Right isosceles triangle (both legs =) –leg = ½ hypotenuse √2 ≈.707 hypotenuse Use properties of 30°- 60°- 90° triangles –leg opposite 30° = ½ hypotenuse –leg opposite 60° = ½ hypotenuse √3 ≈ 0.866 hypotenuse

4. Vocabulary None new

5. Special Right Triangles 45° 30° 60° Remember Pythagorean Theorem a 2 + b 2 = c 2 x x x√2 Pythagorean Theorem a 2 + b 2 = c 2 x 2 + x 2 = (x√2) 2 2x 2 = 2x 2 y 2y y√3 Pythagorean Theorem a 2 + b 2 = c 2 y 2 + (y√3) 2 = (2y) 2 y 2 + 3y 2 = 4y 2

6. Special Right Triangles 45° 30° 60° Side opposite 60° is ½ the hypotenuse times √3 ½ hyp ½ hyp √2 ½ hyp √3 Side opposite 30° is ½ the hypotenuse Side opposite 45° is ½ the hypotenuse times √2

7. Example 1 WALLPAPER TILING The wallpaper in the figure can be divided into four equal square quadrants so that each square contains 8 triangles. What is the area of one of the squares if the hypotenuse of each 45°-45°-90° triangle measures millimeters?

8. Example 1 cont The length of the hypotenuse of one 45°-45°-90° triangle is millimeters. The length of the hypotenuse is times as long as a leg. So, the length of each leg is 7 millimeters. The area of one of these triangles is or 24.5 millimeters. Answer: Since there are 8 of these triangles in one square quadrant, the area of one of these squares is 8(24.5) or 196 mm 2.

9. Example 2 WALLPAPER TILING If each 45°-45°-90° triangle in the figure has a hypotenuse of millimeters, what is the perimeter of the entire square? Answer: 80 mm

10. Example 3 Find a. The length of the hypotenuse of a 45°-45°-90° triangle is times as long as a leg of the triangle. Multiply. Divide. Rationalize the denominator. Divide each side by Answer:

11. Example 4 Find b. Answer:

12. Example 5 Find QR. is the longer leg, is the shorter leg, and is the hypotenuse. Multiply each side by 2. Answer:

13. Example 6 Find BC. Answer: BC = 8 in.

14. Quiz 1 Need-to-Know Arithmetic Mean (AM) or average: (a + b) / 2 Geometric Mean (GM): √ab Altitude = GM of divided hypotenuse Pythagorean Theorem: a 2 + b 2 = c 2 Pythagorean Triples: Whole numbers that solve the theorem Side opposite 30° angle is ½ the hypotenuse Side opposite 45° angle is ½ the hypotenuse times √2 Side opposite 60° angle is ½ the hypotenuse times √3

15. Summary & Homework Summary: –In a 45°- 45°- 90° triangle (isosceles right ∆), the hypotenuse is √2 times the length of the leg. The measures are x, x, and x√2 –In a 30°- 60°- 90° triangle, the measures of the sides are x, x√3, and 2x. Homework: –pg 360, 4-6, 12-17, 21-23