Transparency 7 3 5 Minute Check on Lesson

Transparency 7 -3 5 -Minute Check on Lesson 7 -2 Find x. √ 95 ≈ 9. 7 x 1. 12 √ 1613 ≈ 40. 2 x 2. 7 13 38 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. Yes, Q Determine whether each set of measures forms a right triangle and state whether they form a Pythagorean triple. 4. 16, 30, 33 6. No, No 5 3 13 5. --- , ---8 2 8 Yes, No Which of the following are not the lengths of sides of a right triangle? Standardized Test Practice: A 25, 20, 15 B 4, 7. 5, 8. 5 C 0. 7, 2. 4, 2. 5 Click the mouse button or press the Space Bar to display the answers. D 36, 48, 62

Lesson 7 -3 Special Case Right Triangles

Objectives • Use properties of 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

Vocabulary • None new

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

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

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°-90° triangle measures millimeters?

Example 1 cont The length of the hypotenuse of one 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.

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

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

Example 4 Find b. Answer:

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

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

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 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

Summary & Homework • Summary: – In a 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 2 x. • Homework: – pg 360, 4 -6, 12 -17, 21 -23