# Triangle ABC is a right triangle. If side AC=7 and side BC=10, what is the measure of side AB?

May 15, 2018

It's not clear which one's the hypotenuse so either $\setminus \sqrt{{7}^{2} + {10}^{2}} = \sqrt{149}$ or $\sqrt{{10}^{2} - {7}^{2}} = \sqrt{51}$.

May 15, 2018

It depends on who is the hypothenuse

#### Explanation:

If $A C$ and $B C$ are both legs, then $A B$ is the hypothenuse, and you have

$\setminus {\overline{A B}}^{2} = \setminus {\overline{B C}}^{2} + \setminus {\overline{A C}}^{2}$

from which you deduce

$\setminus \overline{A B} = \sqrt{\setminus {\overline{B C}}^{2} + \setminus {\overline{A C}}^{2}} = \sqrt{100 + 49} = \sqrt{149}$

If, instead, $B C$ is the hypoyhenuse, you have

$\setminus \overline{A B} = \sqrt{\setminus {\overline{B C}}^{2} - \setminus {\overline{A C}}^{2}} = \sqrt{100 - 49} = \sqrt{51}$

May 15, 2018

Depending on which is the right angle, either $\sqrt{51}$ or $\sqrt{149}$

#### Explanation:

Using Pythagoras, ($h y p o t e n u s {e}^{2} = A r {m}^{2} + A r {m}^{2}$)

If BC is the hypotenuse,
$100 = 49 + A {B}^{2}$
$A B = \sqrt{51}$ (length must be positive)

However, if AB is the hypotenuse, then
$A {B}^{2} = 100 + 49$
$A B = \sqrt{149}$ (length must be positive)

AC cannot be the hypotenuse as it is shorter than BC.