04/20/23 5.5: Special Right Triangles

5.5: Special Right Triangles and Areas of Regular

PolygonsExpectations:

G1.2.4: Prove and use the relationships among the side lengths and the angles of 30º- 60º- 90º

triangles and 45º- 45º- 90º triangles.

G1.5.1: Know and use subdivision or circumscription methods to find areas of polygons

G1.5.2: Know, justify and use formulas for the perimeter and area of a regular n- gon.

ACT Prep

If one diagonal of a rhombus is 12 inches long and the other is 32 inches long, how many inches long, to the nearest hundredth of an inch, is a side of the rhombus?

A.8.54

B.17.09

C.34.17

D.35.78

E.48.00

04/20/23 5.5: Special Right Triangles

04/20/23 5.5: Special Right Triangles

If a square has area of x2 square units, what is the length of one of its diagonals?

04/20/23 5.5: Special Right Triangles

45-45-90 Right Triangle Theorem

If a leg of a 45-45-90 right triangle is x units long, then the hypotenuse is x√2 units long.

04/20/23 5.5: Special Right Triangles

30-60-90 Right Triangles

a. sketch an equilateral triangle with sides of 2x units long.b. draw an altitude of the triangle.c. label all known measures.d. what is the length of the altitude?

04/20/23 5.5: Special Right Triangles

30-60-90 Right Triangle Theorem

 In a 30-60-90 right triangle, if the length of the shorter leg is x units, then the longer leg is x√3units and the hypotenuse is 2x units long.

04/20/23 5.5: Special Right Triangles

The hypotenuse of a 30-60-90 right triangle is 20 cm. What are the lengths of the other 2 sides?

What is the perimeter of a 30-60-90 right triangle if the length of the hypotenuse is 8 mm?

04/20/23 5.5: Special Right Triangles

ACT Prep

If the length of a diagonal of a square is 18 inches long, what is the area of the square, in square inches?

A. 9√2

B. 36√2

C. 72

D. 162

E. 324

04/20/23 5.5: Special Right Triangles

ACT Prep

If the length of each side of a regular hexagon is 10 centimeters long, what is the area of the hexagon, to the nearest centimeter?

A.25√3

B.60

C.100√3

D.150√3

E.600√3

04/20/23 5.5: Special Right Triangles

04/20/23 5.5: Special Right Triangles

Center of a Regular Polygon

The center of a regular polygon is the point which is equidistant from the vertices of the regular polygon.

04/20/23 5.5: Special Right Triangles

Apothem of a regular polygon

An apothem of a regular polygon is a segment with one endpoint at the center of the regular polygon and the other endpoint on the polygon, such that the segment is perpendicular to a side of the polygon.

04/20/23 5.5: Special Right Triangles

Center and apothem of a regular polygon

Center of the regular octagon

Apothem of the regular octagon

04/20/23 5.5: Special Right Triangles

Area of a Regular Polygon

1. Locate the center of the regular polygon.

2. Triangulate the polygon using the center as a common vertex.

3. What type of triangles are formed?

4. Draw the altitudes of the triangles.

04/20/23 5.5: Special Right Triangles

Area of a Regular Polygon

5.What are the altitudes in terms of the polygon?

6. What is the area of one triangle?

7. What is the area of the regular polygon expressed as a product?

8. Change to using the perimeter.

04/20/23 5.5: Special Right Triangles

Area of a Regular Polygon Theorem

If a regular polygon has area of A square units, perimeter of p units and an apothem of a units, then

A =

04/20/23 5.5: Special Right Triangles

Assignment

pages 336-338, numbers 10-17(all), 22-38(evens), 44, 45