Carcass wrote:
In the figure, ABCD is a rectangle, and F and E are points on AB and BC, respectively. The area of ∆DFB is 9 and the area of ∆BED is 24. What is the perimeter of the rectangle?
(A) 18
(B) 23
(C) 30
(D) 42
(E) 48
This is a nice question for seeing the different triangles inscribed in shapes.
We need to find the dimensions of the rectangle in order to find its perimeter.So take a look at
∆DFB and
∆BED. Notice that the height of
∆DFB is the length
BC, and the height of
∆BED is
DC.
Now notice that the base of
∆DFB = 2, and we're given that the area of
∆DFB = 9. So with those two pieces of information, we can find the height:
\frac{(b*h)}{2} = 9\frac{(2*h)}{2} = 9The two's cancel and we get:
h = 9So now we have the width of the rectangle, BC, which is 9.Using the same strategy we used above for
∆DFB,
∆BED has a base of 4 and we're given that its area is 24.
\frac{(b*h)}{2} = 24\frac{(4*h)}{2} = 24(2*h) = 24h = 12And now we know the length of the rectangle, DC, which is 12.To finish up, we add all the sides together to find the perimeter:
9+9+12+12 = 42So D is the answer.