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8-7 Special Right Triangles

Warm-Up The diagram at the right is a square with a side length of 3. Identify all the different-size squares, tell how many squares there are of each size, and tell the area of each square. 8-7 Special Right Triangles. 8-7 Special Right Triangles.

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8-7 Special Right Triangles

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  1. Warm-UpThe diagram at the right is a squarewith a side length of 3. Identify all thedifferent-size squares, tell how manysquares there are of each size, and tellthe area of each square.

  2. 8-7 Special Right Triangles

  3. 8-7 Special Right Triangles • The lengths of the sides of right triangles with 30° or 45° angles are related in simple ways.

  4. 8-7 Special Right Triangles • The 45°-45°-90° Triangle This triangle is an isosceles right triangle. Use the Pythagorean Theorem to calculate the length of the hypotenuse. a2 + b2 = c2 x2 + x2 = c2 2x2 = c2 xÖ2 = c

  5. 8-7 Special Right Triangles • Isosceles Right Triangle Theorem In an isosceles right triangle, if a leg has length x, then the hypotenuse has length xÖ2.

  6. 8-7 Special Right Triangles Find the length of XY. According to the Isosceles Right Triangle Theorem, since a leg is the hypotenuse is

  7. 8-7 Special Right Triangles Find the length of PR. According to the Isosceles Right Triangle Theorem,

  8. 8-7 Special Right Triangles • Isosceles Right Triangle Theorem (ammended) In an isosceles right triangle, • if a leg has length x, then the hypotenuse has length xÖ2; • if the hypotenuse has a length of x, then a leg has length (x/2)Ö2.

  9. 8-7 Special Right Triangles Find the area of figure ABCD.

  10. 8-7 Special Right Triangles Find the area of figure ABCD. • Find area of DABC. ½ 8 * 8 = 32ft2 • Find length of DC. AC = 8Ö2, so DC = (8Ö2) Ö2= 8*2=16ft • Find area of DACD. ½ 16 * 8 = 64ft2 • Add areas of DABC and DACD. 32ft2 + 64ft2 = 96ft2

  11. 8-7 Special Right Triangles • The 30°-60°-90° Triangle This triangle is half an equilateral triangle. If the length of each side of the original equilateral triangle is 2x, then the hypotenuse of DABC is 2x and the short leg of DABC is x. Using the Pythagorean Theorem, the long leg of DABC is xÖ3.

  12. 8-7 Special Right Triangles • 30°-60°-90° Triangle Theorem In a 30°-60°-90°triangle, if the length of the shorter leg is x, then the length of the longer leg is xÖ3 and the length of the hypotenuse is 2x.

  13. 4 in 25 ft 6 m Short Long Hypotenuse Leg leg 4 in Ö3 in 8 in 12.5 ft 12.5Ö3 in 25 in 2Ö3 m 6 m 4Ö3 m 8-7 Special Right TrianglesFind the length of the missing sides in the following 30°-60°-90° Triangles.

  14. 8-7 Special Right Triangles Finding the Areas of Regular Polygons Refer to the three regular polygons drawn at the right. (One drawing is incomplete.) Step 1 - If you draw segments joining the center of the regular polygon to its vertices, you create triangles. Explain how you know that these triangles are isosceles. Step 2 - Explain how you know that a is the height of each of these isosceles triangles. Step 3 - In terms of s and a, what is the area of each isosceles triangle? Step 4 - Write a formula for the area of any regular n-gon in terms of s, a, and n. Step 5 - Write a formula for the perimeter p of any regular n-gon with side length s. Step 6 - Use your answers to Steps 3 and 4 to create a formula for the area of any regular polygon in terms of a and p.

  15. 8-7 Special Right Triangles • Regular Polygon Area Theorem The area of a regular polygon is half the product of the length of the apothem a and its perimeter p. A = ½ ap

  16. 8-7 Special Right Triangles In an equilateral triangle, the apothem is √3 . • What are some other names for a regular triangle? • Find the area of the triangle.

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