Question 285786: A rectangle is inscribed in a circle and the ratio between its width and length is 3:4.What is the ratio between the area of the rectangle and the area of the circle?
A. 12 : 25π B. 12 : 25 C. 48 : 25 π D. 25 : 48 E. None of these.
Answer by Grinnell(63)   (Show Source):
You can put this solution on YOUR website! If the ratio between the width and the length is 3:4,
then we represent the sides by 3x and 4x. (It does not matter about the actual length, we just know the ratio is 3 to 4.) Now the area is then going to be represented by 12x^2.
Now we know the area of the rectangle is 12x^2 which will be in the numerator of our ratio answer.
(12x^2 over something!)
Now the rectangle is inscribed in a circle, meaning that its vertices touch the circumfrence of the circle.
It is now that we draw a picture!
Do you see that the diameter of the circle is the diagonal of the rectangle?
What do we know about diagonals of rectangles? The point of intersection bisects the diagonals.
But wait, do you also see the right triangle that we have formed? By the pythaoren (spelling, oops) theorem
we know that a^2 plus b^2 equals c^2. (I assume you know this!)
So, we have determined that the Diameter of the circle is the same as the Hypotenuse of the triangle formed. THIS IS 5X.
Now the area of a circle is pi(r^2) r is 5x/2, then.
We just substitute...(5x/2)^2 (pi) =area of circle.
this equals 25x^2 (pi)/2 (you do the math.)
NOW THIS IS IN THE DENOMINATOR OF OUR RATIO ANSWER!!!
Write out the fraction:
12x^2/25x^2 (pi)/4 Do the math we get...4/25x^2 (pi) TIMES 12x^2, the x^2 cancel out
We are left with 48 over 25(pi)
The answer is C.
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