2019 AMC 10A Problems/Problem 7

Revision as of 19:42, 12 February 2019 by Clara32356 (talk | contribs) (Solution 4)
The following problem is from both the 2019 AMC 10A #7 and 2019 AMC 12A #5, so both problems redirect to this page.

Problem

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 $x+y=10  ?$

$\textbf{(A) } 4 \qquad\textbf{(B) } 4\sqrt{2} \qquad\textbf{(C) } 6 \qquad\textbf{(D) } 8 \qquad\textbf{(E) } 6\sqrt{2}$

Solution 1

The two lines are $y=2x-2$ and $y = \frac{x}{2}+1$, which intersect the third line at $(4,6)$ and $(6,4)$. So we have an isosceles triangle with base $2\sqrt{2}$ and height $3\sqrt{2} \implies \boxed{\textbf{(C) }6}$.

Solution 2

Like in Solution 1, let's first calculate the slope-intercept form of all three lines: $(x,y)=(2,2)$ and $y=x/2 + b$ implies $2=2/2 +b=1+b$ so $b=1$, while $y=2x + c$ implies $2= 2*2+c=4+c$ so $c=-2$. Also, $x+y=10$ implies $y=-x+10$. Thus the lines are $y=x/2 +1, y=2x-2,$ and $y=-x+10$. Now we find the intersections between each of the lines with $y=-x+10$, which are $(6,4)$ and $(4,6)$. Applying the Shoelace Theorem, we can find that the solution is $6 \implies \boxed{\textbf{(C) }6}.$

Solution 3

Like the other solutions, solve the systems to see that the triangles two other points are at $(4, 6)$ and $(6, 4)$. The apply Heron's Formula. The semi-perimeter will be $s = \sqrt{2} + \sqrt{20}$. The area then reduces nicely to a difference of squares, making it $6 \implies \boxed{\textbf{(C) }6}.$

Solution 4

Using the methods above to find the coordinates, we get $(4, 6)$, $(6, 4)$, and $(2,2)$. Using the Shoelace Theorem, we find that it is equal to $(4\cdot 2)-(2\cdot 6)+(2\cdot 4)-(6\cdot 2)+(6 \cdot 6)-(4\cdot 4) = 12$. Because the Shoelace Theorem tells us to find the half of that sum, we get $\boxed{\text{6 (C)}}$

--Claire (clara32356)

See Also

2019 AMC 10A (ProblemsAnswer KeyResources)
Preceded by
Problem 6
Followed by
Problem 8
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 10 Problems and Solutions
2019 AMC 12A (ProblemsAnswer KeyResources)
Preceded by
Problem 4
Followed by
Problem 6
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 12 Problems and Solutions

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