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5-Minute Check on Lesson 7-2

Transparency 7-3. 5-Minute Check on Lesson 7-2. Find x. 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.

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5-Minute Check on Lesson 7-2

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  1. Transparency 7-3 5-Minute Check on Lesson 7-2 • Find x. • 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? √95 ≈ 9.7 √1613 ≈ 40.2 x x 13 7 12 38 Yes, Q 5 3 13 --- , --- , ---- 8 2 8 No, No Yes, No Standardized Test Practice: 36, 48, 62 25, 20, 15 4, 7.5, 8.5 0.7, 2.4, 2.5 D A B C D Click the mouse button or press the Space Bar to display the answers.

  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. 45° 60° x√2 y√3 x 2y y 30° 45° x Pythagorean Theorem a2 + b2 = c2 y2 + (y√3)2 = (2y)2 y2 + 3y2 = 4y2 Pythagorean Theorem a2 + b2 = c2 x2 + x2 = (x√2)2 2x2 = 2x2 Special Right Triangles Remember Pythagorean Theorem a2 + b2 = c2

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

  7. 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? Example 1

  8. 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. Example 1 cont 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 mm2.

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

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

  11. Answer: Example 4 Find b.

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

  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: a2 + b2 = c2 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

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