Triangles ABC and DEF are similar right triangles. If the hypotenuse

Question

Triangles ABC 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 of ABC is the 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.

vitaliyGMAT · Accepted Answer

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.

Nunuboy1994 · Answer

St 1

not enough info about lengths of DEF - could be isosceles right triangle or something else

insuff

St 2

no info about the ABC

insuff

Only a 3 4 5 triangle fits the criteria

C

Aviral1995 · Answer

How do you know that here length of sides of rt. triangle is decreasing? Is it because of ratio we got??

Fdambro294 · Answer

Triangle ABC:

Shorter Leg = a
Other Leg = b
Hypotenuse = c

Triangle DEF:

Shorter Leg = d
Other Leg = e
Hypotenuse = f = 20

The 2 triangles are similar right triangles. Therefore, if we knew the Ratio between the Side Lengths, we could determine triangle ABC’s hypotenuse.

What’s is the length of triangle ABC’s hypotenuse = c = ?

S1:

Perimeter of DEF / Area DEF = Area ABC / Perimeter ABC

(d + e + 20) / (.5de) = (.5ab) / (a + b + c)

Since there are no integer constraints or other original conditions, we can not find unique values for d, e, a, b, and c.

S1 Not sufficient

S2: given that one of the Legs of Triangle DEF is 12 and the Hypotenuse is 20, we can determine all the unique characteristics of triangle DEF.

The side lengths of DEF are in the Pythagorean Triplet Ratio: 3 - 4 - 5

f = 5 (4) = 20

d = 3 (4) = 12

thus, Other Leg = e = 4x———> will be in the same 3-4-5 ratio in which the unknown multiplier is x = 4

e = 4 (4) = 16

Perimeter = 12 + 16 + 20 = 48

Area = (1/2) * (12) * (16) = 96

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

However we do not have any more information about triangle ABC. Since the triangle is a Similar Triangle, we know that the Side Lengths must also be in Pythagorean Triplet Ratio of: 3x : 4x : 5x

But we do not know the Scaling for ABC with just statement 2’s information and can not determine a unique value for c = Hypotenuse of Triangle ABC

S2 not sufficient.

(1) + (2)

1 / 2 = (.5ab) / (a + b + c)

Since S1 provides us with a relationship between the 2 Similar triangles that can helps us determine the Scaling Ratio and S2 tells us everything we need to know about Similar Triangle DEF ——-> together the 2 statements are SUFFICIENT

Let the Scaling Ratio for the side lengths of Triangle ABC = m

Then:

a = 3m

b = 4m

c = 5m

The question asks us to find the Value of the Hypotenuse = c = 5m = ?

(1/2) = (.5 * 3m * 4m) / (3m + 4m + 5m)

(1/2) = (6 * (m)^2) / (12m)

Since m = positive value representing the scaling ratio for a positive Triangle Side length, we can cancel the Power of 2 in the NUM

(1/2) = (6m) / (12)

(1/2) = (1m) / (2)

m = 1 ———> the scaling ratio is m = 1

Hypotenuse = c = 5m = 5(1) = 5

5 is the unique answer

C

Posted from my mobile device

Rickooreo · Answer

gmatophobia   Bunuel

We are not given information that the sides of the triangle are integer, thus can't there be a possibilty of sides of ABC being in roots and satisfy the condition.

gmatophobia · Answer

Rickooreo

Agree that the sides need not be integer, however as the triangles are similar the sides of ABC and the sides of DEF must maintain the same ratio. From Statement 2, we have the information on the sides of DEF, and using the information in Statement 1, we can find the sides.

There are two constraints that the sides of ABC must meet - 1) Must maintain the same ratio 2) Must satisfy Pythagoras theorem. Hence the sides will be unique.

Triangles ABC and DEF are similar right triangles. If the hypotenuse

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Triangles ABC and DEF are similar right triangles. If the hypotenuse

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