Problem page the longer leg of a right triangle is 1ft longer than the shorter leg. the hypotenuse is 9ft longer than the shorter leg. find the side lengths of the triangle.

Hello,

To solve this problem we want to use the Pythagorean Theorem. 
The pythagorean theorem states that for a 90° triangle, 

[tex] a^{2} + b^{2} = c^{2} [/tex]

where a and b represent the two legs of the triangle, and c represents the hypotenuse. 

Let a = the longer leg and b = the shorter leg.
If the longer leg of the triangle is 1 foot longer than the shorter leg, then
a = b +1. 

If the hypotenuse is 9 feet longer than the shorter leg, then c = b + 9.
Using the equations we created, we can plug them into the Pythagorean Theorem to solve for a, b, and c. 

Doing this, we have:
[tex] a^{2} + b^{2} = c^{2} [/tex]
[tex] (b+1)^{2} + b = (b+9)^{2} [/tex]

Expanding this, we get [tex] b^{2} + 2b + 1 + b^{2} = b^{2} + 18b + 81 2b^{2} + 2b + 1 = b^{2} + 18b + 81 b^{2} + 1 = 16b + 81 b^{2} = 16b + 80 b^{2} - 16b - 80 = 0 [/tex]

Solving for b, we get b = 20, and b = -4.
The length of the side of a triangle cannot be negative, so we know that b = 20. 

However, we should check this with the original question to make sure it checks out.

a = b + 1
a = 20 + 1 = 21

c = b + 9
c = 20 + 9 = 29

So, we have a = 21, b = 20, and c = 29. (Also, 20-21-29 is a well known Pythagorean triple)
Using the Pythagorean Theorem, we have:

[tex] 21^{2} + 20^{2} = 29^{2} [/tex]
441 + 400 = 841
841 = 841, checks out.

So, the shorter leg is 20 feet, the longer leg is 21 feet, and the hypotenuse is 29 feet. 

Hope this helps!