Question

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 $x + y = 10?$

$4$

$4\sqrt{2}$

$6$

$8$

$6\sqrt{2}$

Answer: C

Concepts:coordinate geometry triangle area shoelace formula

Difficulty rating: 1280

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.

2019 AMC 12A Problem 5

Answer: C

Difficulty rating: 1280

Solution:

Problem 5 in Other Years