sat-dog

Choice A is correct. For the right triangle shown, the lengths of the legs are $a$ units and $6$ units, and the length of the hypotenuse is $21$ units. The Pythagorean theorem states that in a right triangle, the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse. Therefore, $a^2+6^2={21}^2$. Subtracting $6^2$ from both sides of this equation yields $a^2={21}^2-6^2$. Taking the square root of both sides of this equation yields $a=\pm \sqrt{{21}^2-6^2}$. Since $a$ is a length, $a$ must be positive. Therefore, $a=\sqrt{{21}^2-6^2}$. Thus, for the triangle shown, $\sqrt{{21}^2-6^2}$ represents the value of $a$.

Choice B is incorrect. For the triangle shown, this expression represents the value of $a^2$, not $a$.

Choice C is incorrect and may result from conceptual errors.

Choice D is incorrect and may result from conceptual errors.