2019 AMC 12A Problem 5

Problem 5 of 25EasierGeometry

Two lines with slopes

\dfrac{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?

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Solution

The two lines are

y = \tfrac{1}{2}x + 1

and

y = 2x - 2.

Intersecting each with

x + y = 10

gives the points

(6, 4)

and

(4, 6).

The triangle has vertices

(2, 2),

(6, 4),

and

(4, 6).

By the shoelace formula,

\begin{aligned} &\small \tfrac{1}{2}\left| 2(4 - 6) + 6(6 - 2) + 4(2 - 4) \right| \\ &= \tfrac{1}{2}\left| -4 + 24 - 8 \right| \\ &= 6. \end{aligned}

Thus, the correct answer is C.