2013 AMC 12B Problems/Problem 8

Problem

Line $l_1$ has equation $3x - 2y = 1$ and goes through $A = (-1, -2)$ . Line $l_2$ has equation $y = 1$ and meets line $l_1$ at point $B$ . Line $l_3$ has positive slope, goes through point $A$ , and meets $l_2$ at point $C$ . The area of $\triangle ABC$ is $3$ . What is the slope of $l_3$ ?

$\textbf{(A)}\ \frac{2}{3} \qquad \textbf{(B)}\ \frac{3}{4} \qquad \textbf{(C)}\ 1 \qquad \textbf{(D)}\ \frac{4}{3} \qquad \textbf{(E)}\ \frac{3}{2}$

Solution 1

Line $l_1$ has the equation $y=3x/2-1/2$ when rearranged. Substituting $1$ for $y$ , we find that line $l_2$ will meet this line at point $(1,1)$ , which is point $B$ . We call $\overline{BC}$ the base and the altitude from $A$ to the line connecting $B$ and $C$ , $y=-1$ , the height. The altitude has length $|-2-1|=3$ , and the area of $\triangle{ABC}=3$ . Since $A={bh}/2$ , $b=2$ . Because $l_3$ has positive slope, it will meet $l_2$ to the right of $B$ , and the point $2$ to the right of $B$ is $(3,1)$ . $l_3$ passes through $(-1,-2)$ and $(3,1)$ , and thus has slope $\frac{|1-(-2)|}{|3-(-1)|}=$ $\boxed{\textbf{(B) }\frac{3}{4}}$ .

Solution 2 - Shoelace Theorem

We know lines $l_1$ and $l_2$ intersect at $B$ , so we can solve for that point: $3x-2y=1$ Because $y = 1$ we have: $3x-2(1) = 1$ $3x-2=1$ $3x=3$ $x = 1$

Thus we have $B = (1,1)$ .

We know that the area of the triangle is $3$ , so by Shoelace Theorem we have:

$$ (Error compiling LaTeX. Unknown error_msg)A = \dfrac{1}{2} |(-2x+y-1) - (-2+x-y)|$<cmath>A = \dfrac{1}{2} |-2x+y-1+2-x+y|</cmath> <cmath>3 = \dfrac{1}{2} |-2x+y-1+2-x+y|</cmath> <cmath>6 = |-3x+2y+1|.</cmath>

Thus we have two options:

or

Now we must just find a point that satisfies$ (Error compiling LaTeX. Unknown error_msg)m_{l_3}$is positive.

Doing some guess-and-check yields, from the second equation:

so a valid point here is$ (Error compiling LaTeX. Unknown error_msg)(3,1) $. When calculated, the slope of$ l_3 $in this situation yields$ \boxed{\dfrac{3}{4}}$.

Video Solution

https://youtu.be/a-3CAo4CoWc

~Punxsutawney Phil or sugar_rush

2013 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 7	Followed by Problem 9
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2013 AMC 12B Problems/Problem 8

Contents

Problem

Solution 1

Solution 2 - Shoelace Theorem

Video Solution

See also