Question

How do you do this question?

Answer 1

All the triangles in the figure are similar, so the ratio of the long side to the short side is the same for all.

... 2AC=3AB means AC=(3/2)AB . . . . . the long side is 3/2 times the short side

and

... (2/3)AC=AB . . . . . . . . . . . . . . . . . . . . the short side is 2/3 times the long side

If the square has side 1, then that length is the short side of ΔDEC, which means length DC = 3/2.

Similarly, the side of the square is the long side of ∆FBE, so FB=2/3.

Now, we know the side lengths of ∆ABC, so we can figure its area.

... Area ∆ABC = (1/2)(AB)·(AC)

... = (1/2)(1 + 2/3)·(1 + 3/2) = (1/2)·(5/3)·(5/2)

... Area ∆ABC = 25/12

Since the square has a side length of 1, its area is 1² = 1, and the ratio of the square area to the triangle area is

... 1/(25/12) = 12/25 . . . . . corresponds to selection D

Answer 2

Remark

Nice little problem. There is an awful lot of geometry in this question; there might be a quicker way of doing it, but this is the only way I could find easily.

Step One

Label the sides AB as 2 in length and AC as 3 in length. Be careful that you understand why they are labeled that way. It is because AC has to be shorter than AB. If you call AB = 1 then AC is 1.5. To make the numbers easier to use, I multiplied both of them by 2 so that AB = 2 and AC = 3

Step Two

Call the square side = s. The trick is to show that triangle BFE is similar to triangle CDE. That is the key to the entire problem.

FE is parallel to AD and by extension FE is parallel to AE.

CB is a transversal across these two parallel lines. Therefore <FEB = <C

In the same way <B = <DEC By AA the two triangles are similar.

The proportion of the triangles can be set up

ED/BF = DC/FE

Givens

ED = s [it's the side of a square]

BF = 2 - s

DC = 3 - s

FE = s

s/(2 - s) = (3 - s) / s Study this awhile. Cross Multiply

s^2 = (2 - s)(3 - s) Remove the brackets.

s^2 = 6 - 2s - 3s + s^2 Subtract s^2 from both sides. Combine on the right.

0 = 6 - 5s Subtract 6 from both sides.

-6 = - 5s Divide by minus 5

-6/-5 = s

s = 1.2

The side of the square is 1.2

Area of the square = s^2 = 1.2^2 = 1.44

Step Three

Find the area of the large triangle.

Area = 1/2 AC * AB

Area = 1/2 * 3 * 2

Area = 3

Step Four

Compare the areas.

Area of square / Area of Triangle = 1.44 / 3 = 0.48

Step five

Change to a fraction

0.48 * 100 / 100 = 48 / 100 Divide top and bottom by 4

12 / 25 is your answer which is D

D <<<<< Answer