Question 1

&#10;Use the drawing provided to explain the following theorem.&#10;&#34;The area A of a regular polygon whose apothem has length a and whose perimeter&#10;is P is given by   .&#34;&#10;Given: Regular polygon   with center O and length s for each side;&#10;apothem   so that   Prove:

Accepted Answer

From center O, we draw radii

,

, and

. Because the radii are congruent to each other and the sides of the regular polygon are all congruent to each other as well,

,

, and

are all congruent to each other by SSS.
Each of the congruent triangles has an altitude length of

. Further, the length of each
base of a triangle is s, the length of side of the polygon. Therefore, the area of the regular polygon is

Because the sum of the sides equals perimeter P, we have

.

Question 2

&#10;Where   is the degree measure for the arc of a sector of a circle, the ratio of the area of the sector to that of the area of the circle is given by   . Use this ratio to explain why&#10;the area of the sector is given by   .&#10;[Note: In the figure, the sector with arc measure   is bounded by radii   ,   , and   .]

Accepted Answer

Given that   , it follows that   . Where r is the length of radius of the circle, the area of the circle is given by   . By substitution, it follows that   .

Question 3

&#10;Using the drawing provided and fact that the area of a parallelogram is given by   , show that the area of a triangle is given by   .

Accepted Answer

For the parallelogram (

) with base length b and altitude length h, the area is given by

. By the Area-Addition Postulate,

. But diagonal

of

separates the parallelogram into 2 congruent triangles that have equal areas.
By substitution,

and by division (or multiplication),

.

Question 4

Consider a circle with diameter length d, radius length r, and circumference C. Given that   , explain why the formula for the circumference of a circle is given by   .

Accepted Answer

The answer of Consider a circle with diameter length d,...

Question 5

&#10;Use the drawing provided to explain the following theorem.&#10;&#34;The area of any quadrilateral with perpendicular diagonals of lengths   and   is given by   .&#34;&#10;Given: Quadrilateral   with   at point F;   and   Prove:

Accepted Answer

The answer of  &#10;Use the drawing provided to explain...

Deck 8: Areas of Polygons and Circles