Triangle question

[shonen]

In a certain right triangle, the sum of the lengths of the
two legs and the hypotenuse is 60 inches. If the hypotenuse
is 26 inches, which of the following is the length of one of
the legs?
A. 24 inches
B. 34 inches
C. 29 inches
D. 16 inches
E. 13 inches

The answer is A per the solution below, just wondering if there's a faster method to solve the problem? -

The Pythagorean theorem states that the sum of the squares of the lengths of the legs of a triangle is equal to the square of the length of the hypotenuse.

Let A and B be the lengths of the sides. Let C be the length of the hypotenuse. Thus, we can set up the following equation:

A2 + B2 = C2

We are told that the sum of the legs and hypotenuse is 60 inches. We are also told that the hypotenuse is 26 inches. Thus, we can set up the following equation:

A + B + 26 = 60
A + B = 34
A = 34 - B

Plugging the value of A and C into the first equation:

(34 - B)2 + B2 = 262
(34 - B)(34 - B) + B2 = 676
1156 - 68B + B2 + B2 = 676
2B2 - 68B + 1156 = 676
2B2 - 68B + 480 = 0
(2B - 48)(B - 10) = 0
B = 10 or B = 24

[kssshah]

since there are 2 variables, you need two equations to solve the problem. So i doubt there is an easier method. The only other easier method which reduces some of the calc is to backsolve. start putting values (and fortunately the 1st one works in this example) and you should be able to solve in under 2 mins.

You are looking for 2 #s whose values add up to 34 and squares add up to 26^2 i.e. 676. 24 and 10 will do it. One more pointer, you can just square the last digit of the choice and 34-the choice, add it and see if it is even worth it. for e.g. in above example, choice D is 16. if u square it last digit is 6 (from 36). the counterpart is 18 (34-16). square of last digit is 4 (from 64), 6+4 = 0 <> 6 (from 676).

[fromcttoupenn]

I'm not sure if you find this convenient but this is how I tried.
a+b+c=60
a+b=34
a^2+b^2=(a+b)^2-2ab
1156-2ab

But a^2+b^2=c^2,
Hence 1156-2ab=676
-> ab= 240.
Although you'll get the answer when you try the first option, you can eliminate 16 as 16+15 does not equal 34

[sonikamadala]

I suggest to use the back solving method to solve this question.
we told that a+b+c=60 and c= 26 => a+b = 34
we know that a^2+b^2=26^2

we can eliminate B and C easily as the sum of the square of B or C excceds 26^2. Now left with A, D and E.

Note : unit digit of 26^2 is 6.
Subtitute answer choices for a.
b^2=26^2 - a^2

if a= 24, then b= 10

unit digit of 24^2 is 6 and 10^2 is 0.

Thus unit digit of 26^2-24^2 is zero. So A wins.

Impliment the same logic for D and E.

We will be able to solve this in 2 mins.

[EdYo]

Well, when we can find a*b= 240 we can rewrite it to a*b=2*2*2*2*3*5 and from there we can check the answer. It gonna be faster.

[touche]

Fastest way....
remember standard rt triangle configurations
this one is 13,12,5
you know 13 *2 is one... others may be 24 and 10
check the options

[cristofer]

On the GMAT, whenever a hypotenuse is a multiple of 13, just expect that the other two legs are going to be multiples of 5 and 12. Also, whenever the hypotenuse is a multiple of 5, expect the legs to multiples of 3 and 4.

The GMAT uses these two triangles (and their scaled versions) so frequently, that doing the math is usually not worth it. Scan the answer choices to see if your suspicions are correct, and click it. 99.99% of the time it will be one of the two special triangles.

[sonikamadala]

Originally Posted by cristofer

On the GMAT, whenever a hypotenuse is a multiple of 13, just expect that the other two legs are going to be multiples of 5 and 12. Also, whenever the hypotenuse is a multiple of 5, expect the legs to multiples of 3 and 4.

The GMAT uses these two triangles (and their scaled versions) so frequently, that doing the math is usually not worth it. Scan the answer choices to see if your suspicions are correct, and click it. 99.99% of the time it will be one of the two special triangles.

Thanks for your nice tip buddy