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Difference between revisions of "2019 AMC 10A Problems/Problem 7"

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

Revision as of 18:51, 9 February 2019

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: $(x,y)=(2,2)$ $y=x/2 + b$ becomes $2=2/2 +b=1+b$ so b=1, $y=2x + c$ becomes $2= 2*2+c=4+c$ so c=-2, and $x+y=10$ becomes $y=-x+10$. ($y=x/2 +1, y=2x-2,$ 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 $6 \implies \boxed{C}.$

See Also

2019 AMC 10A (Problems • Answer Key • Resources)
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 (Problems • Answer Key • Resources)
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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