Question

How do you use the Pythagorean Theorem to find the missing side of the right triangle with the given measures given c is the hypotenuse and we have b=3x,c=7x?

Answer 1

`a=2xsqrt10`

Explanation:

The Pythagorean Theorem states that

`a^2+b^2=c^2`

in a triangle with legs `a,b` and hypotenuse `c`, as you've already described in the problem.

With `b=3x` and `c=7x`, we have the relation:

`a^2+(3x)^2=(7x)^2`

Now, recall that when we have something like `(3x)^2`, we have to square both the `3` and the `x`:

`(3x)^2=3^2*x^2=9x^2`

Similarly, for `(7x)^2`:

`(7x)^2=7^2*x^2=49x^2`

Substituting these back in to the Pythagorean Theorem equation, we see that

`a^2+9x^2=49x^2`

Subtract `9x^2` from both sides of the equation.

`a^2=40x^2`

Take the square root of both sides.

`a=sqrt(40x^2)`

We can rewrite `sqrt(40x^2)` as a product of mostly squared terms in order to simplify. For example, it's important to note that `40=4xx10`.

`a=sqrt4*sqrt(x^2)*sqrt10`

`a=2xsqrt10`

Answer 2

`A=2xsqrt(10)`

Explanation:

Using the principle of proportionality disregard the `x's` for now.
Think if it as working on a triangle that has been reduced in scale but is of the same ratio.

By Pythagoras `A^2+B^2 = C^2`

So `A->sqrt(7^2-3^2)" "=" "sqrt(49-9)`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Technically we could write `A=xsqrt(7^2-3^3)`
It is simpler just to leave it out for now but incorporate it at the end.
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

`A->sqrt(2^2xx10)`

`A->2sqrt(10)`

Scaling back up we have

`A=2xsqrt(10)`