The perimeter of a certain isosceles right triangle

[s.aamershah]

The perimeter of a certain isosceles right triangle is 16 + 16*2^(1/2). What is the length of the hypotenuse of the triangle?

A) 8
B) 16
C) 4*2^(1/2)
D) 8*2^(1/2)
E) 16*2^(1/2)

The correct answer is B.

I'm having trouble solving this problem. The perimeter of a right isosceles triangle is base+height+hypotenuse = x+x+x*2^(1/2) = 16 + 16*2^(1/2)

I square both sides to get rid of the square root and end up with (x^2)+(x^2)+(x^2)*2 = 256+256*2
Factoring that out I get x^2(1+1+1*2) = 768
4*x^2=768
x^2=192
x = approximately 14
(2^1/2) = approximately 1.4
14*1.4 = 19.6

(note: 2^1/2 is square root two)

What am I doing wrong?

[singhvikramveer]

Since this triangle is isosceles it has 2 legs of equal length, with the third leg being the hypotenus. Hypotenuse triangles have the following ratio of sides: x : x : xsqroot2, the last one being the hypotenuse.

Now, look at the sum 16 + 16sqroot2.

Now what is given: Perimeter is 16 + 16sqrt2 assuming the 16sqrt2 as hyp we will get perimeter as 32 + 16sqr2 but that is not the case so, the hyp is 16 and other two sides are 8sqrt2..

[s.aamershah]

Thanks for a the reply. That is a very straightforward way to look at the question!

As a general note, can you or anyone else confirm that when the hypotenuse of a right isosceles triangle is a whole integer (not x sqrroot 2) then the legs are always 1/2 of the hypotenuse sqrroot 2?

[singhvikramveer]

I doubt that. No that is not true you can look at it like a square cut in half through the diagonal. in a square all sides are equal and the diagonal is side.sqrt2 and the ratio in that case is x:xsqrt2... I hope that helps..

[mschwrtz]

s.aamershah is right.

The ratio x:x:sqrt2x (leg:leg:hypotenuse) is in fact equal to the ratio
sqrt2x:sqrt2x:2x (leg:leg:hypotenuse).

But why not just plug into pythagorean theorem and check?

[msbinu]

Hypot of this triangle of side x is x * sqrt 2

x+x+sqrt 2 *x = 16+16*sqrt 2
2x+sqrt 2 * x = 16(1+sqrt 2)

x * sqrt 2(sqrt 2 +1 ) = 16 (1+ sqrt 2 )

x * sqrt 2 = 16

[mschwrtz]

Which gives a total perimeter of 16 + 16sqrt2. Check.