Question

Square ABCD is inscribed in circle P, with a diagonal that is 18 centimeters long. Find the exact length of the apothem of square ABCD.

a. 18√2
b. 9√2
c. 9√2 over 2
d. 9 over 2

Answer 1

`d=a √(2) \\ \\ a √(2) =18 \\ \\ a= (18)/( √(2) ) = (18 √(2) )/(2)=9 √(2) \\ \\ x= (a)/(2) \\ \\ x= (9 √(2) )/(2)`

Answer 2

C

Explanation:

The Square ABCD inscribed in a circle is shown in the picture attached.

• The diagonal is broken down into two 9 cm parts as shown.

• We let
  `y` be the length of the apothem [line from center of square to midpoint of side].

• Also, we let
  `x` be half the length of the side of the square.
• Since square,
  `x` and
  `y` are equal

Let's find the side length of the square using pythagorean theorem:

`(Side Length)^(2)+(Side Length)^(2)=18^(2)\\2SideLength^(2)=324\\SideLength^(2)=162\\SideLength=√(162)`

Since,
`x` is HALF of SIDE LENGTH,
`x` is:

`x=(√(162))/(2)\\x=(√(81)*√(2))/(2)\\x=(9√(2))/(2)`

Since,
`x` and
`y` are equal, we can say
`y=(9√(2))/(2)`

Apothem's length is
`(9√(2))/(2)`. Answer choice C is right.