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8-7 Special Right Triangles

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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.

8-7 Special Right Triangles

- The lengths of the sides of right triangles with 30° or 45° angles are related in simple ways.

- 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

- Isosceles Right Triangle Theorem
In an isosceles right triangle, if a leg has length x, then the hypotenuse has length xÖ2.

Find the length of XY.

According to the Isosceles Right Triangle Theorem, since a leg is

the hypotenuse is

Find the length of PR.

According to the Isosceles Right Triangle Theorem,

- 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.

Find the area of figure ABCD.

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

- 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.

- 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.

4 in

25 ft

6 m

Short LongHypotenuse

Leg leg

4 in Ö3 in8 in

12.5 ft 12.5Ö3 in25 in

2Ö3 m 6 m4Ö3 m

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.

- 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

In an equilateral triangle, the apothem is √3 .

- What are some other names for a regular triangle?
- Find the area of the triangle.