2019 AMC 10A Problem 7

Question

Two lines with slopes

\frac{1}{2}

and

2

intersect at

(2, 2).

What is the area of the triangle enclosed by these two lines and the line

x + y = 10?

4

4\sqrt{2}

6

8

6\sqrt{2}

Find all three pairwise intersections of the lines

Use the distance or shoelace formula for the resulting triangle

The two lines through

(2,2)

have equations

\begin{aligned} y&=\frac12x+1,\\ y&=2x-2. \end{aligned}

Their intersections with

x+y=10

are

(6,4)

and

(4,6)

, respectively.

Thus the vertices are

(2,2),(6,4),

and

(4,6)

. The segment joining the last two points has length

2\sqrt2

, its midpoint is

(5,5)

, and the distance from

(2,2)

to that midpoint is

3\sqrt2

.

The area is therefore

\frac12(2\sqrt2)(3\sqrt2)=6.

Thus, C is the correct answer.

Problem 7 in other years