The circle below has a radius of $5$ cm. The figure inscribed in the circle is a rectangle with width $2x$. Express each of the following in terms of $x$.

1. The perimeter of the rectangle.

   Step 1(a):

   Draw a diagonal. If the radius is 5, how long is the diagonal?

   Step 2(a):

   Use the Pythagorean Theorem to find the length L.

   $(2x)^2+L^2=10^2$

   $L^2=100-4x^2$

   $L=\sqrt{100-4x^2}$

   $L=\sqrt{4(25-x^2)}$

   $L=2\sqrt{25-x^2}$

   Step 3(a):

   Perimeter = 2(width) + 2(length)

   Answer:

   $\text{Perimeter}=2(2x)+2(2\sqrt{25-x^2})$

   $\text{Perimeter}=4x+4\sqrt{25-x^2}$

2. The area of the rectangle.

   Hint (b):

   Area = (Length)(Width)

   $\text{Area}=(2x)(2\sqrt{25-x^2})$

   $\text{Area}=4x\sqrt{25-x^2}$

   Answer (b):