using pythagorean theorem to get a polynomial equationA right triangle has a hypotenuse that is 13 inches long. If one leg is 2 inches more than twice the other leg, find the length of the longer...

using pythagorean theorem to get a polynomial equation

A right triangle has a hypotenuse that is 13 inches long. If one leg is 2 inches more than twice the other leg, find the length of the longer leg of the right triangle.

Asked on by kaia

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litteacher8's profile pic

litteacher8 | High School Teacher | (Level 3) Distinguished Educator

Posted on

I agree. The Pythagorean Theorem is one of the oldest and most important ones. Basically, it says that the squares of both of the shorter legs of the triangle added together will be equal to the square of the longest side, the hypotenuse. So just square each side, add the, together, and take the square root to get the hypotenuse. If you have one of the other sides missing, work backwards.
pohnpei397's profile pic

pohnpei397 | College Teacher | (Level 3) Distinguished Educator

Posted on

You know that your legs are x and 2x + 2.  You know your hypotenuse is 13.  You know that the Pythagorean Theorem says that the sum of the squares of the legs is equal to the square of the hypotenuse.

So now you know that

x^2 + (2x + 2)^2 = 169

So now you need to do

(2x + 2)*(2x +2) so as to find the square of the second leg.  That gets you

4x^2 + 8x + 4.

So now your equation is

x^2 + 4x^2 + 8x + 4 = 169

5x^2 + 8x - 165 = 0

Now you have a quadratic equation and can solve for x.

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

The Pythagorean theorem is joining th sum of the squares of the cathetus of a right angle triangle and the square of hypotenuse.

a^2 = b^2 + c^2

a = 13 inches = hypotenuse

b = one cathetus

c = the other cathetus = 2 + 2b

We'll substitute the values of the sides of the triangle in Pythagorean theorem:

13^2 = b^2 + (2+ 2b)^2

We'll raise to square and we'll get:

169 = b^2 + 4 + 8b + 4b^2

We'll combine like terms and we'll get:

5b^2 + 8b + 4 - 169 = 0

5b^2 + 8b - 165 = 0

b1 = [-8 + sqrt(64 + 3300)]/10

b1 = (-8+58)/10

b1 = 5

b2 = -66/10

Since a length of a side cannot be negative, we'll keep just the positive value of b = 5 inches long.

The length of the longer leg is:

c = 2 + 2b

c = 2 + 10

c = 12 inches long

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