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7.4 Special Right Triangles

7.4 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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7.4 Special Right Triangles

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  1. 7.4 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°∆ A Theorem 7.8 In a 45°- 45°- 90° triangle, the length of the hypotenuse is √2 times the length of a leg. hypotenuse = √2 • leg OR The legs and hypotenuse have a ratio of 1:1:√2. x√2 45 ° 45 ° B C

  5. The length of the hypotenuse of a 45°- 45°- 90° triangle is times as long as a leg of the triangle. Example 1: Find a.

  6. Divide each side by Answer: Example 1: Rationalize the denominator. Multiply. Divide.

  7. The length of the hypotenuse of a 45°- 45°- 90° triangle is times as long as a leg of the triangle. Example 2: P Find q. 45° q 7 45° Q R 7 PR is the hypotenuse so PR = q = 7√2 . q = 7√2 Answer:

  8. Answer: Your Turn: Find b.

  9. 30° - 60° - 90°∆ A Theorem 7.9 • 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 ° Be sure you realize the shorter leg is opposite the 30°& the longer leg is opposite the 60°. 30 ° B x√3 C Hypotenuse = 2 ∙ shorter legLonger leg = √3 ∙ shorter leg OR The legs and hypotenuse have a ratio of 1:√3:2.

  10. Example 3: Find QR.

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

  12. Example 4: Find PR.

  13. is the longer leg, is the shorter leg, and is the hypotenuse. Example 4: PR = √3 • (4 √3 / 3) PR = 4√9 / 3 PR = 4 • 3 / 3 PR = 4 Answer: PR = 4

  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 and in Quadrant III we know that it lies on a vertical gridline below X. First, 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:

  19. Assignment • Geometry: Workbook Pgs. 133 – 135 #1 – 25

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