In a certain right triangle, the hypotenuse is 9 cm longer than the first leg and 2 cm longer than the second. Find the length of the hypotenuse of the triangle

Question

solve and type an equation

Answer 1

Represent the length of the hypotenuse by "H"

Represent the length of the first leg by "H-9"

Represent the length of the second leg by "H-2"

Substitute into the pythagorean theorem

H∧2=(H-9)∧2+(H-2)∧2

Solve for H

H∧2=H∧2-18H+81+H∧2-4H+4

Simplify

H∧2=2H∧2-22h+85

0=H∧2-22H+85

Factor the quadratic

0=(H-5)(H-17)

set H-5=0  and   H-17=0

H=5   or H=17

Answer: H=17 (based upon the parameters defined in the problem H≠5 since the hypotenuse is 9 cm longer than one leg and thus would render the length of that leg as a negative number)

 

Answer 2

Use the Pythagorean theorem to solve this. You know that in a right triangle, a²+b²=c², where c is the hypotenuse and a and b are the other sides. In this case, your hypotenuse (c) is 9 cm longer than one side, so that side is 9 cm shorter than the hypotenuse. Let's call that side a; you know that a = c - 9. The other side is 2 cm shorter than the hypotenuse - let's call it b - so b = c - 2.

 

So far we have:

 

a = c-9
b = c-2
c

 

So:

 

a² = (c-9)² = c²-18c+81 

b² = (c-2)^{2 =} c²-4c+4

 

Since a²+b²=c2,  (c²-18c+81) + (c²-4c+4) = c²

 

 

Combine terms to get the quadratic: c²-22c-85-0.
This can be factored into (c-17)(c-5)=0, so c can equal 17 or 5.

 

Since the hypotenuse is 9 cm longer than another side, it cannot equal 5 as this would result in a leg of negative length.

 

So, the hypotenuse must be 17, and the other sides are therefore 15cm and 8cm. (To check your math, make sure that 15²+8²=172, which is true.

Answer 3

let x be the length of the first leg

then x+9 is the length of the hypothesis and x+(9-2) = x+7 is the length of the second leg

 

by Pythagorean Theorem  x^2 + (x+7)^2 = (x + 9)^2

 

expanding x^2 + X^2 + 14x + 49 = X^2 +18x + 81

 

combining like terms and setting equal to zero

x^2 -4x -32 = 0

 this factors to 

(x -8) (x+4) = 0

 

by the zero property x = -4 or x = 8

 we throw out x = -4 because the length must be positive

 

and the first leg length = 8

second leg = 8+7= 15

Hypotenuse = 8+9 = 17

 

check 8^2 + 15^2 = 17^2 = 289