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Special Right Triangles

Special Right Triangles. Objectives. Use properties of 45 ° - 45° - 90° triangles Use properties of 30° - 60° - 90° triangles. Side Lengths of Special Right ∆s.

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Special Right Triangles

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  1. Special Right Triangles

  2. Objectives • Use properties of 45° - 45° - 90° triangles • Use properties of 30° - 60° - 90° triangles

  3. Side Lengths of Special Right ∆s • Right triangles whose angle measures are 45° - 45° - 90°or 30° - 60° - 90°are called special right triangles. The theorems that describe the relationships between the side lengths of each of these special right triangles are as follows:

  4. 45° - 45° - 90°∆ Theorem 7.6 In a 45°- 45°- 90° triangle, the length of the hypotenuse is √2 times the length of a leg. hypotenuse = √2 • leg x√2 45 ° 45 °

  5. 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?

  6. 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: 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.

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

  8. The length of the hypotenuse of a 45°- 45°- 90° triangle is times as long as a leg of the triangle. Example 2: Find a.

  9. Divide each side by Answer: Example 2: Rationalize the denominator. Multiply. Divide.

  10. Answer: Your Turn: Find b.

  11. 30° - 60° - 90°∆ Be sure you realize the shorter leg is opposite the 30°& the longer leg is opposite the 60°. Theorem 7.7 • In a 30°- 60°- 90° triangle, the length of the hypotenuse is twice as long as the shorter leg, and the length of the longer leg is √3 times as long as the shorter leg. 60 ° 30 ° x√3 Hypotenuse = 2 ∙ shorter legLonger leg = √3 ∙ shorter leg

  12. Example 3: Find QR.

  13. is the longer leg, is the shorter leg, and is the hypotenuse. Answer: Example 3: Multiply each side by 2.

  14. Your Turn: Find BC. Answer: BC = 8 in.

  15. COORDINATE GEOMETRY is a30°-60°-90° triangle with right angle X and as the longer leg. Graph points X(-2, 7) and Y(-7, 7), and locate point W in Quadrant III. Example 4:

  16. Graph X and Y. lies on a horizontal gridline of the coordinate plane. Since will be perpendicular to it lies on a vertical gridline. Find the length of Example 4:

  17. is the shorter leg. is the longer leg. So, Use XY to find WX. Point W has the same x-coordinate as X.W is located units below X. Answer: The coordinates of W are or about Example 4:

  18. COORDINATE GEOMETRY is at 30°-60°-90° triangle with right angle R and as the longer leg. Graph points T(3, 3) and R(3, 6) and locate point S in Quadrant III. Answer: The coordinates of S are or about Your Turn:

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