Find the Missing Leg AB in Right Triangle ABC with Hypotenuse 9 and Base 6

To find the length of side $AB$ in the right triangle $\triangle ABC$, we will use the Pythagorean Theorem. The theorem is given by:

$a^2 + b^2 = c^2$

In this equation, $c$ is the hypotenuse, and $a$ and $b$ are the other two sides. Based on the given information, we have:

• $AC = 6$ is one leg.
• $BC = 9$ is the hypotenuse.
• $AB$ is the unknown leg we want to find.

Substituting the known values into the Pythagorean Theorem:

$AB^2 + 6^2 = 9^2$

Simplifying the equation:

$AB^2 + 36 = 81$

Subtract 36 from both sides to solve for $AB^2$:

$AB^2 = 81 - 36$

$AB^2 = 45$

To find $AB$, take the square root of both sides:

$AB = \sqrt{45}$

Therefore, the length of side $AB$ is $\sqrt{45}$.