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Difference between revisions of "2016 AMC 8 Problems/Problem 2"

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== Problem ==
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In rectangle <math>ABCD</math>, <math>AB=6</math> and <math>AD=8</math>. Point <math>M</math> is the midpoint of <math>\overline{AD}</math>. What is the area of <math>\triangle AMC</math>?
 
In rectangle <math>ABCD</math>, <math>AB=6</math> and <math>AD=8</math>. Point <math>M</math> is the midpoint of <math>\overline{AD}</math>. What is the area of <math>\triangle AMC</math>?
  
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<math>\textbf{(A) }12\qquad\textbf{(B) }15\qquad\textbf{(C) }18\qquad\textbf{(D) }20\qquad \textbf{(E) }24</math>
 
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==Solution==
 
 
<asy>draw((0,4)--(0,0)--(6,0)--(6,8)--(0,8)--(0,4)--(6,8)--(0,0));
 
<asy>draw((0,4)--(0,0)--(6,0)--(6,8)--(0,8)--(0,4)--(6,8)--(0,0));
 
label("$A$", (0,0), SW);
 
label("$A$", (0,0), SW);
Line 12: Line 11:
 
label("$4$", (0, 2), W);
 
label("$4$", (0, 2), W);
 
label("$6$", (3, 0), S);</asy>
 
label("$6$", (3, 0), S);</asy>
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 +
<math>\textbf{(A) }12\qquad\textbf{(B) }15\qquad\textbf{(C) }18\qquad\textbf{(D) }20\qquad \textbf{(E) }24</math>
 +
 +
==Solutions==
  
 
===Solution 1===
 
===Solution 1===
  
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Use the triangle area formula for triangles: <math>A = \frac{bh}{2},</math> where <math>A</math> is the area, <math>b</math> is the base, and <math>h</math> is the height. This equation gives us <math>A = \frac{4 \cdot 6}{2} = \frac{24}{2} =\boxed{\textbf{(A) } 12}</math>.
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Using the triangle area formula for triangles: <math>A = \frac{bh}{2},</math> where <math>A</math> is the area, <math>b</math> is the base, and <math>h</math> is the height. This equation gives us <math>A = \frac{4 \cdot 6}{2} = \frac{24}{2} =\boxed{\textbf{(A) } 12}</math>.
  
 
===Solution 2===
 
===Solution 2===
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A triangle with the same height and base as a rectangle is half of the rectangle's area.  This means that a triangle with half of the base of the rectangle and also the same height means its area is one quarter of the rectangle's area.  Therefore, we get <math>\frac{48}{4} =\boxed{\textbf{(A) } 12}</math>.
 
A triangle with the same height and base as a rectangle is half of the rectangle's area.  This means that a triangle with half of the base of the rectangle and also the same height means its area is one quarter of the rectangle's area.  Therefore, we get <math>\frac{48}{4} =\boxed{\textbf{(A) } 12}</math>.
  
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===Solution 3(a check)===
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We can find the area of the entire rectangle, DCBA to be <math>8 \cdot 6=48</math> and find DCM area to be <math>\frac{6 \cdot 4}{2} = 12</math> and BCA to be <math>\frac{6 \cdot 8}{2}=24</math>, <math>48-12-24=</math> <math>\boxed{\textbf{(A) } 12}</math>.
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===Solution 4===
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A triangle is half of a rectangle. So since M is the midpoint of DA, we can see that triangle DAC is half of the whole rectangle. And it is also true that triangle MAC is half of triangle DAC, so triangle MAC would just be 1/4 of the whole rectangle. Since the rectangle's area is 48, 1/4 of 48 would be 12. Which gives us the answer <math>\boxed{\textbf{(A) } 12}</math>.
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===Solution 5 (complicated way)===
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We can subtract the total areas of triangles DCM and ABC from the rectangle ABCD. For triangle DCM, the base is 4 and the height is 6, so we multiply 4 and 6, then divide by 2 to get 12. For triangle ABC, the base is 4 and the height is 8, so we multiply 4 and 8, then divide by 2 to get 24. We add 24 and 12 to get 36. Then, we calculate the area of rectangle ABCD, which is 48. We subtract 36 from 48, resulting in <math>\boxed{\textbf{(A) } 12}</math>.
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-BananaBall00
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 +
==Video Solution==
 +
 +
https://youtu.be/PoMegS49Efs?si=pxq8rn_1peOb-ZCq
 +
 +
A solution so simple a 12-year-old made it!
 +
 +
~Elijahman~
 +
 +
==Video Solution (THINKING CREATIVELY!!!)==
 +
https://youtu.be/BQAztEkvYNw
 +
 +
~Education, the Study of Everything
 +
 +
==Video Solution==
 +
https://www.youtube.com/watch?v=LqnQQcUVJmA (has questions 1-5)
 +
 +
https://youtu.be/tHzYXORdbzQ
 +
 +
~savannahsolver
 +
 +
==See Also==
 
{{AMC8 box|year=2016|num-b=1|num-a=3}}
 
{{AMC8 box|year=2016|num-b=1|num-a=3}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 10:38, 20 June 2024

Problem

In rectangle $ABCD$, $AB=6$ and $AD=8$. Point $M$ is the midpoint of $\overline{AD}$. What is the area of $\triangle AMC$?

[asy]draw((0,4)--(0,0)--(6,0)--(6,8)--(0,8)--(0,4)--(6,8)--(0,0)); label("$A$", (0,0), SW); label("$B$", (6, 0), SE); label("$C$", (6,8), NE); label("$D$", (0, 8), NW); label("$M$", (0, 4), W); label("$4$", (0, 2), W); label("$6$", (3, 0), S);[/asy]

$\textbf{(A) }12\qquad\textbf{(B) }15\qquad\textbf{(C) }18\qquad\textbf{(D) }20\qquad \textbf{(E) }24$

Solutions

Solution 1

Using the triangle area formula for triangles: $A = \frac{bh}{2},$ where $A$ is the area, $b$ is the base, and $h$ is the height. This equation gives us $A = \frac{4 \cdot 6}{2} = \frac{24}{2} =\boxed{\textbf{(A) } 12}$.

Solution 2

A triangle with the same height and base as a rectangle is half of the rectangle's area. This means that a triangle with half of the base of the rectangle and also the same height means its area is one quarter of the rectangle's area. Therefore, we get $\frac{48}{4} =\boxed{\textbf{(A) } 12}$.

Solution 3(a check)

We can find the area of the entire rectangle, DCBA to be $8 \cdot 6=48$ and find DCM area to be $\frac{6 \cdot 4}{2} = 12$ and BCA to be $\frac{6 \cdot 8}{2}=24$, $48-12-24=$ $\boxed{\textbf{(A) } 12}$.

Solution 4

A triangle is half of a rectangle. So since M is the midpoint of DA, we can see that triangle DAC is half of the whole rectangle. And it is also true that triangle MAC is half of triangle DAC, so triangle MAC would just be 1/4 of the whole rectangle. Since the rectangle's area is 48, 1/4 of 48 would be 12. Which gives us the answer $\boxed{\textbf{(A) } 12}$.

Solution 5 (complicated way)

We can subtract the total areas of triangles DCM and ABC from the rectangle ABCD. For triangle DCM, the base is 4 and the height is 6, so we multiply 4 and 6, then divide by 2 to get 12. For triangle ABC, the base is 4 and the height is 8, so we multiply 4 and 8, then divide by 2 to get 24. We add 24 and 12 to get 36. Then, we calculate the area of rectangle ABCD, which is 48. We subtract 36 from 48, resulting in $\boxed{\textbf{(A) } 12}$.

-BananaBall00

Video Solution

https://youtu.be/PoMegS49Efs?si=pxq8rn_1peOb-ZCq

A solution so simple a 12-year-old made it!

~Elijahman~

Video Solution (THINKING CREATIVELY!!!)

https://youtu.be/BQAztEkvYNw

~Education, the Study of Everything

Video Solution

https://www.youtube.com/watch?v=LqnQQcUVJmA (has questions 1-5)

https://youtu.be/tHzYXORdbzQ

~savannahsolver

See Also

2016 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 1 		Followed by
Problem 3
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 AJHSME/AMC 8 Problems and Solutions

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