Pythagorean theorem

Question

Hi, I am studying for a test and here is a practice question. I have not done this type of math for a long time and youtube videos are not helping right now. I have the question and answer, but I cannot seem to figure out the reasoning with the explanation they give to me. Help with simple steps would be greatly appreciated! 

 

RST is a right triangle, what is the value of x?

 

Leg 1 is X

Leg 2 is 2x+2

Hypotenuse is x+8

 

I understand a2 plus b2 = c2.  x2+(2x+2)2=(x+8)2

 

I get a little confused "expanding the problem" and then solving from there. Please help!

 

Answer 1

Let's pick it up at:    x²+(2x+2)²=(x+8)²

 

Actually, you have:     x²+ (2x+2)(2x+2) = (x+8)(x+8)

 

The method of expanding the squares is like expanding two polynomials -- it's called F-O-I-L for First-Outside-Inside-Last.

 

Using FOIL, we get:

                     x² +     4x² + 4x + 4x + 4   =  x² + 8x + 8x + 64

                                4x²  - 8x - 60 = 0             (collect all term on left by subtracting)

                                   x²  - 2x - 15 = 0            (divide everything by 4 to make life easier)

                                (x-5)(x+3) = 0             (factor or use quadratic equation)

                                 x = 5   or  x = -3             (but length of 3 doesn't make sense)

                                  x = 5

 

Checking (very important):

    Is  5*5 + 12*12 = 13*13    ?

          25 + 144   = 169    ?     yes!

           

Answer 2

x² + (2x + 2)² = (x + 8)²

If we expand everything,

x² + 4x² + 8x + 4 = x2 + 16x + 64

4x² - 8x - 60 = 0

x² - 2x -15 = 0

 

Factoring this,

(x - 5)(x + 3) = 0

x = 5, -3.

But, since we can't have a negative length,

x = 5

 

Checking this out by substituting back into the original equation,

 

5²  + (2 * 5 + 2)² = (5 + 8)²

 

25 + 12² = 13²

25 + 144 = 169

Answer 3

Remember, Pythagorean Theorem says  for a right triangle with legs a,b and hypotenuse c that a²+b²=c²

 

in this case a=x, b=2x+2, c=x+8

Plugging into the formula we have

x²+(2x+2)²=(x+8)²

x²+4x²+8x+4=x²+16x+64

We can turn this into a quadrtic equation by 

moving everything to one side and combining like terms

4x²-8x-60=0

factor out a 4

4(x²-2x-15)=0

divide both sides by 4

x²-2x-15=0

We can factor this.  The factors of -15 that add to -2 are -5,3

x²-2x-15=(x-5)(x+3) =0

Either x-5 =0      or    x+3=0

           x=5        or     x=-3

 

Since a leg will not have. negative value, we throw out -3

 

X= 5

 

check:  b= 2x+2= 12,   c=x+8=13

a²+b²=c² 

5² + 12² = 13²

25+144=169

169=169

our answer works.