Question

The $y$ -intercepts, $P$ and $Q,$ of two perpendicular lines intersecting at the point $A(6,8)$ have a sum of zero. What is the area of \triangleriangle APQ?

$45$

$48$

$54$

$60$

$72$

Solution:

We have that the $y$ -intercepts are an equal distance from the origin since their values sum to $0.$

Let this distance be $z.$ We also have that that the distance from $A$ to the origin is $z$ since it is the median to the midpoint of the hypotenuse.

We then know that $z = \sqrt{6^2 + 8^2} = 10$ by the distance formula. We know the altitude from $A$ to $\overline{PQ}$ is $6$ (it is just the $x$ -value of $A$ ).

We also know that $PQ = 2 \cdot 10 = 20,$ which tells us that the area $[APQ] = \dfrac{1}{2} \cdot 6 \cdot 20 = 60.$

Thus, D is the correct answer.

2014 AMC 10A Problem 14

Solution:

Problem 14 in Other Years