One leg of a right triangle is 7cm less than the length of the other leg. The length of the hypotenuse is 13?

Question

One leg of a right triangle is 7cm less than the length of the other leg. The length of the hypotenuse is 13 cm. What is the lengths of the legs?

Answer 1

legs 12 and 5 and hypotenuse 13.

12:5:13 is one of those "common" right triangles (like 3:4:5) that you should have memorized, because it appears on standardized tests all the time.

However, you can also solve it rigorously with the pythagorean theorem:

a^2 + b^2 = c^2

You know that b=(a-7) and c=13, so substitute:

a^2 + (a-7)^2 = 13^2
2a^2 -14a + 49 = 169
2a^2 - 14a - 120 = 0
a^2 - 7a - 60 = 0 (divide by two)
(a-12)(a+5) = 0 (factor)
a = 12 or a = -5

Discounting the negative measure (-5), the answer is a=12. Knowing that, b = (a-7) = (12-7) = 5.

Answer 2

Let's label the longer leg x. So the shorter leg is x-7. By the Pythagorean theorem (a^2 + b^2 = c^2),

x^2 + (x-7)^2 = 13^2
x^2 + (x-7)(x-7) = 169
x^2 + x^2 -14x + 49 = 169
2x^2 - 14x - 120 = 0
x^2 - 7x - 60 = 0
(x - 12)(x + 5) = 0
x = 12 or x = -5

-5 doesn't make any sense as a length, so 12 is the the length of the longer leg. The shorter leg has length 12 - 7 = 5.

Answer 3

Let a=smallest length=x-7
Let b=middle length=x
Let c=hypotenuse=13cm

(x-7)^2 +x^2 = 13^2
(x-7)(x-7) + x^2 = 169
x^2-14x+49+x^2=169
2x^2-14x-120 = 0
2(x^2-7x-60)=0
2(x-12)(x+5)=0
x-12 = 0 or x+5 = 0
x=12 or x=-5
Since we are talking about lenghts of something, x=12
x-7 = 5

So the lengths are 5,12, 13
Lets see if it works
5^2+12^=13^2
25+144=169
169=169, so we are correct.