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

Revision as of 23:28, 26 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

Let's first work out the slope-intercept form of all three lines: $(x,y)=(2,2)$ and $y=\frac{x}{2} + b$ implies $2=\frac{2}{2} +b=1+b$ so $b=1$ , while $y=2x + c$ implies $2= 2 \cdot 2+c=4+c$ so $c=-2$ . Also, $x+y=10$ implies $y=-x+10$ . Thus the lines are $y=\frac{x}{2} +1, y=2x-2,$ and $y=-x+10$ . Now we find the intersection points between each of the lines with $y=-x+10$ , which are $(6,4)$ and $(4,6)$ . Using the distance formula and then the Pythagorean Theorem, we see that we have an isosceles triangle with base $2\sqrt{2}$ and height $3\sqrt{2}$ , whose area is $\boxed{\textbf{(C) }6}$ .

Solution 2

Like in Solution 1, we determine the coordinates of the three vertices of the triangle. Now, using the Shoelace Theorem, we can directly find that the area is $\boxed{\textbf{(C) }6}$ .

Solution 3

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

Solution 4

Like in the other solutions, we find, either using algebra or simply by drawing the lines on squared paper, that the three points of intersection are $(2, 2)$ , $(4, 6)$ and $(6, 4)$ . We can now draw the bounding square with vertices $(2, 2)$ , $(2, 6)$ , $(6, 6)$ and $(6, 2)$ , and deduce that the triangle's area is $16-4-2-4=\boxed{\textbf{(C) }6}$ .

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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

@@ Line 10: / Line 10: @@
 Let's first work out the slope-intercept form of all three lines:
 <math>(x,y)=(2,2)</math> and <math>y=\frac{x}{2} + b</math> implies <math>2=\frac{2}{2} +b=1+b</math> so <math>b=1</math>, while <math>y=2x + c</math> implies <math>2= 2 \cdot 2+c=4+c</math> so <math>c=-2</math>. Also, <math>x+y=10</math> implies <math>y=-x+10</math>. Thus the lines are <math>y=\frac{x}{2} +1, y=2x-2,</math> and <math>y=-x+10</math>.
-Now we find the intersection points between each of the lines with <math>y=-x+10</math>, which are <math>(6,4)</math> and <math>(4,6)</math>. Using the distance formula and then the Pythagorean Theorem, we see that we have an isosceles triangle with base <math>2\sqrt{2}</math> and height <math>3\sqrt{2}</math>, whose area is \boxed{\textbf{(C) }6}<math>.
+Now we find the intersection points between each of the lines with <math>y=-x+10</math>, which are <math>(6,4)</math> and <math>(4,6)</math>. Using the distance formula and then the Pythagorean Theorem, we see that we have an isosceles triangle with base <math>2\sqrt{2}</math> and height <math>3\sqrt{2}</math>, whose area is <math>\boxed{\textbf{(C) }6}</math>.
 ==Solution 2==
-Like in Solution 1, we determine the coordinates of the three vertices of the triangle. Now, using the [[Shoelace Theorem]], we can directly find that the area is </math>\boxed{\textbf{(C) }6}<math>.
+Like in Solution 1, we determine the coordinates of the three vertices of the triangle. Now, using the [[Shoelace Theorem]], we can directly find that the area is <math>\boxed{\textbf{(C) }6}</math>.
 ==Solution 3==
-Like in the other solutions, solve the systems of equations to see that the triangle's two other vertices are at </math>(4, 6)<math> and </math>(6, 4)<math>. Then apply Heron's Formula: the semi-perimeter will be </math>s = \sqrt{2} + \sqrt{20}<math>, so the area reduces nicely to a difference of squares, making it </math>\implies \boxed{\textbf{(C) }6}<math>.
+Like in the other solutions, solve the systems of equations to see that the triangle's two other vertices are at <math>(4, 6)</math> and <math>(6, 4)</math>. Then apply Heron's Formula: the semi-perimeter will be <math>s = \sqrt{2} + \sqrt{20}</math>, so the area reduces nicely to a difference of squares, making it <math>\implies \boxed{\textbf{(C) }6}</math>.
 ==Solution 4==
-Like in the other solutions, we find, either using algebra or simply by drawing the lines on squared paper, that the three points of intersection are </math>(2, 2)<math>, </math>(4, 6)<math> and </math>(6, 4)<math>. We can now draw the bounding square with vertices </math>(2, 2)<math>, </math>(2, 6)<math>, </math>(6, 6)<math> and </math>(6, 2)<math>, and deduce that the triangle's area is </math>16-4-2-4=\boxed{\textbf{(C) }6}$.
+Like in the other solutions, we find, either using algebra or simply by drawing the lines on squared paper, that the three points of intersection are <math>(2, 2)</math>, <math>(4, 6)</math> and <math>(6, 4)</math>. We can now draw the bounding square with vertices <math>(2, 2)</math>, <math>(2, 6)</math>, <math>(6, 6)</math> and <math>(6, 2)</math>, and deduce that the triangle's area is <math>16-4-2-4=\boxed{\textbf{(C) }6}</math>.
 ==See Also==

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