alexpavlos wrote:
A rectangle is inscribed in a circle of radius r. If the rectangle is not a square, which of the following could be the perimeter of the rectangle?
A. 2r sqr3
B. 2r (sqr3 + 1)
C. 4r sqr2
D. 4r sqr3
E. 4r (sqr3 + 1)
Any smart, quick way of solving this one other than brute force?
Since the radius of the circle is r, the diameter must be 2r. Now imagine the rectangle. The diameter must be the hypotenuse of the right angled triangle of the rectangle. Say, if its sides are a and b,
\((2r)^2 = a^2 + b^2 = 4r^2\)
So when you square a and b and sum them, you should get 4r^2
The given options are the perimeter of the rectangle i.e. 2(a+b). So I ignore 2 of the options and try to split the leftover into a and b.
The obvious first choices are options (B) and (E) since we can see that we can split the sum into 3 and 1.
\(a + b = r\sqrt{3} + r\)
Now check:
\((\sqrt{3}r)^2 + r^2 = 4r^2\)
That is what we wanted. Hence, the answer is (B)
Thanks Karishma , but could you please elaborate more ? I still do not understand