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

$d=a \sqrt{2} \\ \\ a \sqrt{2} =18 \\ \\ a= \frac{18}{ \sqrt{2} } = \frac{18 \sqrt{2} }{2}=9 \sqrt{2} \\ \\ x= \frac{a}{2} \\ \\ x= \frac{9 \sqrt{2} }{2}$

Answer 2

Answer:

C

Step-by-step 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:

$(Side Length)^{2}+(Side Length)^{2}=18^{2}\\2SideLength^{2}=324\\SideLength^{2}=162\\SideLength=\sqrt{162}$

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

$x=\frac{\sqrt{162}}{2}\\x=\frac{\sqrt{81}*\sqrt{2}}{2}\\x=\frac{9\sqrt{2}}{2}$

Since, $x$ and $y$ are equal, we can say $y=\frac{9\sqrt{2}}{2}$

Apothem's length is $\frac{9\sqrt{2}}{2}$. Answer choice C is right.