Problem Solving

Rectangle ABCD has a perimeter of 26. The half circle with diameter AD has an area of 8Ï€. What is the perimeter of the part of the figure that is not shaded?

A. 26 + 4Ï€
B. 18 + 8Ï€
C. 18 + 4Ï€
D. 14 + 4Ï€
E. 14 + 2Ï€

The OA is C.

I'm really confused with this PS question. Please, can any expert assist me with it? Thanks in advanced.

Hello LUANDATO.

Let's see your question.

The area of the half circle is 8Ï€, so the complete circle would have an area of 16Ï€. It implies that r=AD/2=4.

So, AD=8.

By the other hand, AB+BC+CD+DA=2AB+2AD=2AB+2*8=2AB + 16 = 26. It implies that 2AB=10, so AB=5.

Now, the lenght of the arc AD is equal to π*r, that is to say, 4π.

Finally, the perimeter is $$AB+BC+CD+arc\left(AD\right)=5+8+5+4\pi=18+4\pi.$$ So, the correct answer is C .

I really hope this explanation may help you.

Feel free to ask me again if you have a doubt.

Regards.