word problem with algebra

Question

The lengths of the sides of a right triangle, a and b, are related to the length of the hypotenuse, c, by the formula a^2+b^2=c^2. In the right triangle shown below, what is the length of the missing side?

 

the vertical side on the left is a, longest slanted side is 10 in, and bottom is 5 in. 

 

Answer 1

The long slanted is c, the hypotenuse, so c=10. The bottom would be b, so b=5. We are solving for a, so filing in the equation with what we have would leave us with:

 

(a)^2 + (5)^2 = (10)^2

 

if we simplify:

 

(a)^2 + 25 = 100

 

We want to get alone, so the next step is to subtract 25 from each side

 

(a)^2 = 75

 

Now to get alone, we take the square root of 75.

 

a= 8.66

Answer 2

"The longest slanted side" is the hypotenuse and it is 10 in.

"bottom is 5 in."  That must be b, because the "vertical side on the left is a."

 

   a^2 + b^2 = c^2

   a^2 + 5^2 = 10^2      (just plugging in values)

   a^2 = 10^2 - 5^2       (subtract 5^2 from both sides)

   a^2 = 100 - 25            (calculate squares)

   a^2 = 75                     (calculate difference)

    a = ± √75                   (find square root)

    a = 5√3                     (only + value makes sense; also take out √25)

 

Special note:

    There are three "very famous" triangles that you will see on homework and on tests.  They are:

 

      angles                 sides

    30-60-90              1*N, SQRT(3)*N, 2*N

    45-45-90               1*N, 1*N, SQRT(2)*N

   (learn later)            3*N, 4*N, 5*N

 

   This problem has sides of 5, SQRT(3)*5, 10         (N=5)