# 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?

Jul 31, 2016

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

${\left(x + 7\right)}^{2} = {35}^{2} + {x}^{2}$ or ${x}^{2} + 14 x + 49 = {35}^{2} + {x}^{2}$ or

$14 x = {35}^{2} - 49 = 1225 - 49 = 1176$ or $x = \frac{1176}{14} = 84$ and sides are

$91$, $35$ and $84$

2 - If $35$ is hypotenuse than

${\left(x + 7\right)}^{2} + {x}^{2} = {35}^{2}$ or ${x}^{2} + 14 x + 49 + {x}^{2} = 1225$ or

$2 {x}^{2} + 14 x - 1176 = 0$ or ${x}^{2} + 7 x - 588 = 0$ and

$x = \frac{- 7 \pm \sqrt{{7}^{2} - 4 \times 1 \times \left(- 588\right)}}{2} = \frac{- 7 \pm \sqrt{49 + 2352}}{2}$

or $x = \frac{- 7 \pm \sqrt{2401}}{2} = \frac{- 7 \pm 49}{2}$

But as using minus sign gives negative answer, which is not possible, only possibility is $x = \frac{- 7 + 49}{2} = \frac{42}{2} = 21$ and sides are $21$, $28$ and $35$.

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

or $21$, $28$ and $35$.