Question

A triangle with vertices $(6, 5),$ $(8, -3),$ and $(9, 1)$ is reflected about the line $x = 8$ to create a second triangle. What is the area of the union of the two triangles?

$9$

$\dfrac{28}{3}$

$10$

$\dfrac{31}{3}$

$\dfrac{32}{3}$

Solution:

The reflected triangle has vertices $(7,1)$ , $(8,-3)$ , and $(10,5)$ .

The line through $(6,5)$ and $(9,1)$ is $y=-\frac43x+13$ , so it meets $x=8$ at $y=\frac73$ . By symmetry, the union is two congruent triangles with vertical base from $(8,-3)$ to $(8,\frac73)$ .

That base has length $\frac73+3=\frac{16}{3}$ , and each triangle has horizontal height $2$ . Hence the union area is $2\left(\frac12\cdot\frac{16}{3}\cdot2\right)=\frac{32}{3}$ .

Thus, E is the correct answer.

2013 AMC 10A Problem 16

Solution:

Problem 16 in Other Years