If a rectangle with length sqrt(a) units and width sqrt(b)

Question

If a rectangle with length  \sqrt{a}  units and width  \sqrt{b}  units is inscribed in a circle of radius 5 units, what is the value of a + b?
 
A. 10
B. 20
C. 25
D. 50
E. 100

Still taking prac exam right now, but this problem was driving me nuts, don't know OA yet, will post when I get done with exam.

The way I see it you can't solve this problem, there's a million different rectangles...I tried researching this on the internet, and everything I've read says there's an infinite number of solutions.

Bunuel · Accepted Answer

A right triangle inscribed in a circle must have its hypotenuse as the diameter of the circle . Since a rectangle is made of two right triangles then a rectangle inscribed in a circle must have its diagonal as the diameter of the circle . Thus (\sqrt{a})^2+(\sqrt{b})^2=10^2 --> a+b=100 . Answer: E. Similar questions to practice: m22-73309-20.html rectangle-abcd-is-inscribed-in-circle-p-what-is-the-area-of-161961.html in-the-xy-coordinate-system-rectangle-abcd-is-inscribed-wit-155992.html a-rectangle-is-inscribed-in-a-circle-of-radius-r-if-the-132684.html Hope this helps.

WoundedTiger · Answer

For a rectangle to be drawn in a circle, the diagonal of the rectangle= diameter of Circle=10 
and in a rectangle of side c and d respectively we have c^2+d^2= l^2 where l is the diagonal...

In this case we side as \sqrt{a} and\sqrt{b} so a+b= 100....

Do share the OA

WholeLottaLove · Answer

Here is the thing though, why can't you have rectangles of different side dimensions inscribed in the circle as in my attached image?

Just for kicks, I drew the attached image out with a pen, paper, a circular template and a ruler. I understand that the radius is 5 (so diameter is 10) but if we drew a rectangle with almost no height it would be just slightly longer while being a lot more narrow. The ratio of length to width wouldn't change enough to make them the same area.

EDIT: Thinking about it, it makes a bit more sense. They're testing to see whether you completely understand the concept of a right triangle. In a right triangle (which this is, it's two of them) a^2 + b^2 MUST EQUAL c^2. Drawing it out throws you off. If you inscribe a rectangle in a circle and you change the dimensions of that rectangle while still keeping it inscribed, the change in width will offset the change in height and vice versa. I guess my hand drawn attempt at understanding this was incorrect. In other words:

a^2 + b^2 = c^2
6^2 + 8^2 = 100
36 + 64 = 100

(a-b)^2 + (b+a)^2 = 100 as well.

Simple, yet very tricky.

NinetyFour · Answer

Bunuel

Is it okay to assume that when we create a hypotenuse for a rectangle we cut it into a 30-60-90 triangle?

Hence  \sqrt{a} = 5  and  \sqrt{b} = 5\sqrt{3} ?

Carlosfernando25 · Answer

The diagonal of the rectangle is 2*R = 2*5=10

Applying the Pythagorean Theorem gives=

10^2 = ( \sqrt{a} )^2 + ( \sqrt{b} )^2

100= a+b

E.

adityasuresh · Answer

Diagonal of rectangle is equal to the diameter of circle.

bumpbot · Answer

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If a rectangle with length sqrt(a) units and width sqrt(b)

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If a rectangle with length sqrt(a) units and width sqrt(b)

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