AAPL wrote: ↑Sat Feb 01, 2020 7:33 am
Manhattan Prep
Triangles ACB and DEF are similar right triangles. if the hypotenuse of triangle DEF has a length of 20, what is the length of the hypotenuse of triangle ABC?
1) The ratio of the perimeter of ABC to the area ABC is reciprocal of the ratio of the perimeter of DEF to the area of DEF.
2) One of the legs in triangle DEF has a length of 12.
OA
C
Let's take each statement one by one.
1) The ratio of the perimeter of ABC to the area ABC is reciprocal of the ratio of the perimeter of DEF to the area of DEF.
Certainly insufficient.
2) One of the legs in triangle DEF has a length of 12.
Certainly insufficient.
(1) and (2) together
For the ∆DEF, we know that hypotenuse EF = 20; say the length of the other leg DF is 12. Thus, ED = length of another leg = √(20^2 – 12^2) = 16
• Perimeter of ∆ DFE = 12 + 16 + 20 = 48;
• Area of ∆ DFE = 1/2*(12*16) = 96;
Even if we had assumed 12 as the length of ED, we would have got the same perimeter and the area.
Thus, the ratio of the area of ∆DEF to its perimeter = 96/48 = 2;
Thus, the ratio of the perimeter of ∆ABC to its area = 2 (given)
Say the two legs of ∆ABC are a and b; thus, the hypotenuse = √(a^2 + b^2)
Since ∆ABC and ∆DEF are similar, a/b = 16/12 = 4/3
Say a = 4x and b = 3x; thus, hypotenuse = √(a^2 + b^2) = 5x
Again, the ratio of the perimeter of ∆ABC to its area = 2 = (4x + 3x + 5x) / (1/2*4x*3x) = 12x/6x^2
=> x = 1.
Thus, the hypotenuse of ∆ ABC = 5x = 5*1 = 5. Sufficient.
The correct answer:
C
Hope this helps!
-Jay
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