giobas wrote:

triangles ABC and DEF are similar right triangles. if the hypotenuse of triangle DEF has a length of 20, what is length of the hypotenuse of triangle ABC?

1) the ratio of perimeter of ABC to the area of the 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 length of 12

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Hi

(1) Perimeter(DEF)/Area(DEF) = Area(ABC)/Perimeter(ABC).

We have only the length of the hypothenuse of DEF. That's not sufficient.

(2) We have both hypothenuse and leg of DEF, hence we can find the other leg, perimeter and area, but no information about scale factor to corresponing sides or areas of another triangle. Insufficient.

(1)&(2) Perimeter(DEF) = 48, Area(DEF) = 96

Perimeter (DEF) / Area (DEF) = 48/96 = 1/2

Perimeter (ABC) / Area (ABC) = 2

With decreasing of lengths of sides of right triangles the area will shrink faster than perimeter, hence our triangle ABC should be smaller in size. Let's check several pythagorian triples:

Initially we have 16, 12 and 20. If we decrease the lengths by 2 this won't influence the pythagorean relationship.

Next: 8^2 + 6^2 = 10^ ----> 64 + 36 = 100. Perimeter/area = (8+6+10) / 1/2*8*6 = 24/24 = 1. We are going in the right direction.

Next well known: 4^2 + 3^2 = 5^2 ---> Perimeter/area = (4 + 3 + 5) / 1/2*4*3 = 12/6 = 2.

Our hypothenuse is 5.

Sufficient.

Answer C.