In the figure above, ABCD is a rectangle inscribed in a circle. If the length of the AB is three times the length of AD, then what is the ratio of the area of the rectangle to the area of the circle (Figure not drawn to scale).
A. 1:2
B. 3:2Ï€
C. 2:5
D. 4:3Ï€
E. 6:5Ï€
The OA is E.
Please, can any expert assist me with this PS question? I don't have it clear and I appreciate if any explain it for me. Thanks.
Hi AAPL,
Let's take a look at your question.
In the rectangle ABCD, if AD = 1 then AB = 3
Area of rectangle = Length * Width
Area of rectangle = 3 * 1 = 3
Area of circle ca be calculated using formula
$$\pi r^2$$
To find the area of circle, we need to find the radius of the circle first.
In Triangle ADC,
$$\left(AC\right)^2=\left(AD\right)^2+\left(DC\right)^2$$
$$\left(AC\right)^2=\left(AD\right)^2+\left(AB\right)^2$$
$$\left(AC\right)^2=\left(1\right)^2+\left(3\right)^2$$
$$\left(AC\right)^2=1+9$$
$$\left(AC\right)^2=10$$
$$AC=\sqrt{10}$$
Since AC is the diameter of the circle, the radius will be half of the length of AC.
$$r=\ \frac{AC}{2}=\frac{\sqrt{10}}{2}$$
Now, area of circle will be:
$$=\pi r^2=\pi\left(\frac{\sqrt{10}}{2}\right)^{^2}$$
$$=\pi\left(\frac{10}{4}\right)$$
$$=\pi\left(\frac{5}{2}\right)$$
Let's find the ratio of the area of rectangle to area of circle.
Area of rectangle : Area of circle
$$3\ :\ \frac{5}{2}\pi$$
$$2\times3\ :\ 2\times\frac{5}{2}\pi$$
$$6\ :\ 5\pi$$
Therefore, Option
E is correct.
Hope it helps.
I am available if you'd like any follow up.
