Using Pythagorean Theorum, sides are 2m+10, 3m, and 5m

Question

a=2m+10, b=3m, c=5m. I cant seem to figure out how to expand past the first step. So far I have (2m+10)²+(3m)² +(5m)² 

I am taking this course online & my professor is of no help. I am terrible at math. Please help.

Answer 1

OK, I will help.

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A little coaching and tutoring before we get to the problem.

Remember a number squared or a number to the second power, is just that number times itself.
So for example   10^2 = 10x10 = 100,  and  N^2 = N*N

But you have to remember to include the parenthesis, so for example:
(3n)^2 = (3n)*(3n) = 3*3*n*n = 9*n^2 or 9n^2   <--- remember we can multiply in any order

You also need the FOIL method if it is (x+k)^2 = (x+k)*(x+k);

FOIL stands for "First, Outside, Inside, Last.

 

Now back to the problem

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(2m + 10)^2 + (3m)^2 = (5m)^2  <--- you were off to a good start; Thank you , Michael J.

 

Write it like this, maybe it will help:

 

(2m + 10)*(2m+10) + (3m)*(3m) = (5m)*(5m)

 

 

Let's do this in 3 parts:

 

 First of all (3m)*(3m), as previously mentioned, is 3*3*m*m = 9*m^2 

 

 Likewise 5m*5m = 5*5*m*m = 25*m^2

 

Now, we have the FOIL:

  (2m + 10)*(2m+10) =     (2m)*(2m) +    <--- FIRST

                                        (2m)*10     +    <--- OUTSIDE

                                         10*(2m)    +    <--- INSIDE

                                         10*10              <--- LAST

 

which simplifies to

 

            2*2*m*m +

             2*10*m +

           10*2*m +

           10*10

 

which then becomes:

     4*m^2 + 20*m + 20*m + 100

 

finally, we combine the two middle cross terms to get:

  4*m^2 + 40*m + 100

 

putting all these pieces together we get:

    4*m^2 + 40*m + 100 + 9*m^2 = 25*m^2

 

  13*m^2 + 40*m + 100 = 25*m^2   <-- combines the like terms on the left side

 

Now we need to move everything to one side.

The highest degree is 2 (quadratic). I like my 

coefficients positive, so the term with exponent 2

that has the smaller coefficient will move. That means,

everything on the left will move to the right, changing

signs in the process. The equation becomes:

 

0 = 25*m^2 - 13*m^2 - 40*m - 100

 

0 = 12*m^2 - 40*m - 100  <--- combines like terms again

 

0 = 3*m^2 - 10*m - 25 <--- divides everything by 4

 

0 = (  3*m     +     5 )(m    -     5  ) <--- factors; this is a problem onto itself; it is FOIL method

                                                              in reverse; the only way to get 3m^2 is 3m*m upon doing "FIRST"

                                                              The only way to get -25 upon doing "LAST is 5 x -5. Trial and error

                                                               will get you this factorization; It's hard to explain how to factor

                                                               in just one step. 

 

3*m + 5 = 0   or m-5 = 0   <--- zero product property; one of them must be zero if their product is zero

 

3*m = -5 or m-5 = 0 <--- solving each one for m

 

m = -5/3  or m = 5 <--- solves for m

 

But if m is negative, then so is 3m and 5m,

and they represent the measures of the the

leg and hypotneuse of the triangle. The side

measures of the triangle cannot be negative,

so this answer is disqualified and rejected.

 

So m=5.

 

Then 2m+10 = 2*5 + 10 = 10 + 10 = 20

3m = 3*5 = 15

5m = 5*5 = 25

 

Left hand side: 20^2 + 15^2 = 400 + 225 = 625

Right hand side: 25*25 = 625

 

So m=5:

The legs of the right triangle are 20 and 15.

The hypotenuse is 25.

The pythagorean checks out to 625.

 

I have done my best to explain each step in horrifying,

graphic detail. It is now YOUR responsibility to peruse and read every

step to make sure YOU understand all of this. Yes, the problem is

long and the factoring is tricky. PLEASE contact us if there is ANYTHING

you do not understand. BE SPECIFIC as to which STEP in the problem 

you do not understand. "THE WHOLE THING" does not do us any good.

Thank you for the problem, Best wishes and good luck.

 

 

 

 

 

Answer 2

(2m + 10)² + (3m)² = (5m)²

 

This is the equation you get applying Pythagorean theorem.  You just solve for m in order to determine the actual side length.