Question

How do you use the Pythagorean Theorem to find the missing side of the right triangle with the given measures: side 1: x + 7, side 2: 35, side 3: x?

Answer 1

Solutions are `91`, `35` and `84`

or `21`, `28` and `35`.

Explanation:

Pythagoras theorem states that in a right angled triangle, square of the hypotenuse the largest side is equal to sum of squares of other two sides.

As the larges side is hypotenuse, either side 1 or side 2 could be hypotenuse. Note side 3 cannot be hypotenuse as side 1 is greater than side 3. Hence, there could be two possibilities.

1 - If `x+7` is hypotenuse than

`(x+7)^2=35^2+x^2` or `x^2+14x+49=35^2+x^2` or

`14x=35^2-49=1225-49=1176` or `x=1176/14=84` and sides are

`91`, `35` and `84`

2 - If `35` is hypotenuse than

`(x+7)^2+x^2=35^2` or `x^2+14x+49+x^2=1225` or

`2x^2+14x-1176=0` or `x^2+7x-588=0` and

`x=(-7+-sqrt(7^2-4xx1xx(-588)))/2=(-7+-sqrt(49+2352))/2`

or `x=(-7+-sqrt2401)/2=(-7+-49)/2`

But as using minus sign gives negative answer, which is not possible, only possibility is `x=(-7+49)/2=42/2=21` and sides are `21`, `28` and `35`.

Hence solutions are `91`, `35` and `84`

or `21`, `28` and `35`.