SOLUTION: find the lengths of each side of the right triangle short leg is 2x+1 long leg is 11x+5 hypotenuse is 12x+1 I know that first you're suppose to write the problem out like this

Algebra ->  Triangles -> SOLUTION: find the lengths of each side of the right triangle short leg is 2x+1 long leg is 11x+5 hypotenuse is 12x+1 I know that first you're suppose to write the problem out like this      Log On


   

Question 841172: find the lengths of each side of the right triangle
short leg is 2x+1
long leg is 11x+5
hypotenuse is 12x+1
I know that first you're suppose to write the problem out like this
(12x+1)^2=(2x+1)^2+(11x+5)^2
but after this I got lost
please help and break it down as much as possible
please and thank you :D
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
So far so good. Now we just need to solve for x.

%2812x%2B1%29%5E2=%282x%2B1%29%5E2%2B%2811x%2B5%29%5E2

%2812x%2B1%29%2812x%2B1%29=%282x%2B1%29%282x%2B1%29%2B%2811x%2B5%29%2811x%2B5%29



144x%5E2%2B12x%2B12x%2B1=4x%5E2%2B2x%2B2x%2B1%2B121x%5E2%2B55x%2B55x%2B25

144x%5E2%2B24x%2B1=125x%5E2%2B114x%2B26

144x%5E2%2B24x%2B1-125x%5E2-114x-26=0

19x%5E2-90x-25=0

Now from here, you could factor, but I like using the quadratic formula (since it always works for any quadratic)

Solved by pluggable solver: Quadratic Formula
Let's use the quadratic formula to solve for x:


Starting with the general quadratic


ax%5E2%2Bbx%2Bc=0


the general solution using the quadratic equation is:


x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29




So lets solve 19%2Ax%5E2-90%2Ax-25=0 ( notice a=19, b=-90, and c=-25)





x+=+%28--90+%2B-+sqrt%28+%28-90%29%5E2-4%2A19%2A-25+%29%29%2F%282%2A19%29 Plug in a=19, b=-90, and c=-25




x+=+%2890+%2B-+sqrt%28+%28-90%29%5E2-4%2A19%2A-25+%29%29%2F%282%2A19%29 Negate -90 to get 90




x+=+%2890+%2B-+sqrt%28+8100-4%2A19%2A-25+%29%29%2F%282%2A19%29 Square -90 to get 8100 (note: remember when you square -90, you must square the negative as well. This is because %28-90%29%5E2=-90%2A-90=8100.)




x+=+%2890+%2B-+sqrt%28+8100%2B1900+%29%29%2F%282%2A19%29 Multiply -4%2A-25%2A19 to get 1900




x+=+%2890+%2B-+sqrt%28+10000+%29%29%2F%282%2A19%29 Combine like terms in the radicand (everything under the square root)




x+=+%2890+%2B-+100%29%2F%282%2A19%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)




x+=+%2890+%2B-+100%29%2F38 Multiply 2 and 19 to get 38


So now the expression breaks down into two parts


x+=+%2890+%2B+100%29%2F38 or x+=+%2890+-+100%29%2F38


Lets look at the first part:


x=%2890+%2B+100%29%2F38


x=190%2F38 Add the terms in the numerator

x=5 Divide


So one answer is

x=5




Now lets look at the second part:


x=%2890+-+100%29%2F38


x=-10%2F38 Subtract the terms in the numerator

x=-5%2F19 Divide


So another answer is

x=-5%2F19


So our solutions are:

x=5 or x=-5%2F19




Now because the hypotenuse is 12x%2B1, this means that when x+=+-5%2F19 the hypotenuse will be 12x%2B1+=+12%28-5%2F19%29%2B1=-2.1578947368421. But we can't have a negative length. So x+=+-5%2F19 cannot be allowed as a solution.

That means x+=+5 is the only solution.

Now plug in x+=+5 into each expression

Short Leg: 2x+%2B+1+=+2%2A5%2B1+=+11

Long Leg: 11x+%2B+5+=+11%2A5%2B5+=+60

Hypotenuse: 12x+%2B+1+=+12%2A5%2B1+=+61

=========================================================

Summary:

Short Leg: 11 units

Long Leg: 60 units

Hypotenuse: 61 units