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Difference between revisions of "2010 AMC 10B Problems/Problem 7"

(Created page with 'The triangle is isosceles. The height of the triangle is therefore given by <math>h = \sqrt{10^2 - ( \dfrac{12}{2})^2} = \sqrt{64} = 8</math> Now, the area of the triangle is <…')
 
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The triangle is isosceles. The height of the triangle is therefore given by <math>h = \sqrt{10^2 - ( \dfrac{12}{2})^2} =  \sqrt{64} = 8</math>
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==Problem==
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A triangle has side lengths <math>10</math>, <math>10</math>, and <math>12</math>. A rectangle has width <math>4</math> and area equal to the
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area of the triangle. What is the perimeter of this rectangle?
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<math>\textbf{(A)}\ 16 \qquad \textbf{(B)}\ 24 \qquad \textbf{(C)}\ 28 \qquad \textbf{(D)}\ 32 \qquad \textbf{(E)}\ 36</math>
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 +
==Solution 1==
 +
The triangle is isosceles. The height of the triangle is therefore given by <math>h = \sqrt{10^2 - \left( \dfrac{12}{2} \right)^2} =  \sqrt{64} = 8</math>
  
 
Now, the area of the triangle is <math>\dfrac{bh}{2} = \dfrac{12*8}{2} = \dfrac{96}{2} = 48 </math>
 
Now, the area of the triangle is <math>\dfrac{bh}{2} = \dfrac{12*8}{2} = \dfrac{96}{2} = 48 </math>
  
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We have that the area of the rectangle is the same as the area of the triangle, namely 48. We also have the width of the rectangle: 4.
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We have that the area of the rectangle is the same as the area of the triangle, namely <math>48</math>. We also have the width of the rectangle: <math>4</math>.
  
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The length of the rectangle therefor is:
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The length of the rectangle therefore is:
 
<math>l = \dfrac{48}{4} = 12</math>
 
<math>l = \dfrac{48}{4} = 12</math>
  
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The answer is:
 
The answer is:
  
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<math> \boxed{\mathrm{(D)} = 32} </math>
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<math> \boxed{\textbf{(D)}\ 32} </math>
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==Solution 2==
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An alternative way to find the area of the triangle is by using Heron's formula, <math>A=\sqrt{(s)(s-a)(s-b)(s-c)}</math> where <math>s</math> is the semi-perimeter of the triangle (meaning half the perimeter). Here, the semi-perimeter is <math>(10+10+12)/2 = 16</math>. Thus the area equals <math>\sqrt{(16)(16-10)(16-10)(16-12)} = \sqrt{16*6*6*4} = 48. </math> We know that the width of the rectangle is <math>4</math>, so <math>48/4 = 12</math>, which is the length. The perimeter of the rectangle is <math>2(4+12) = </math> <math> \boxed{\textbf{(D)}\ 32} </math>.
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(Solution by Flamedragon)
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 +
==Solution 3==
 +
Note that a triangle with side lengths <math>10,10,12</math> is essentially <math>2</math>  “6,8,10” right triangles stuck together. Hence, the height is <math>8</math>, and our area is <math>48</math>.
 +
 
 +
So, the length of the rectangle is <math>\frac{48}{4}=12</math>, and our perimeter <math>P=2(4+12)=\boxed{\textbf{(D)}\ 32}</math>
 +
 
 +
==Video Solution==
 +
https://youtu.be/UGLvhMp67Ag
 +
 
 +
~Education, the Study of Everything
 +
 
 +
=Video Solution==
 +
https://youtu.be/I3yihAO87CE?t=89
 +
 
 +
~IceMatrix
 +
 
 +
==See Also==
 +
{{AMC10 box|year=2010|ab=B|num-b=6|num-a=8}}
 +
{{MAA Notice}}

Latest revision as of 08:00, 1 September 2022

Problem

A triangle has side lengths $10$, $10$, and $12$. A rectangle has width $4$ and area equal to the area of the triangle. What is the perimeter of this rectangle?

$\textbf{(A)}\ 16 \qquad \textbf{(B)}\ 24 \qquad \textbf{(C)}\ 28 \qquad \textbf{(D)}\ 32 \qquad \textbf{(E)}\ 36$

Solution 1

The triangle is isosceles. The height of the triangle is therefore given by $h = \sqrt{10^2 - \left( \dfrac{12}{2} \right)^2} = \sqrt{64} = 8$

Now, the area of the triangle is $\dfrac{bh}{2} = \dfrac{12*8}{2} = \dfrac{96}{2} = 48$

We have that the area of the rectangle is the same as the area of the triangle, namely $48$. We also have the width of the rectangle: $4$.

The length of the rectangle therefore is: $l = \dfrac{48}{4} = 12$

The perimeter of the rectangle then becomes: $2l + 2w = 2*12 + 2*4 = 32$

The answer is:

$\boxed{\textbf{(D)}\ 32}$

Solution 2

An alternative way to find the area of the triangle is by using Heron's formula, $A=\sqrt{(s)(s-a)(s-b)(s-c)}$ where $s$ is the semi-perimeter of the triangle (meaning half the perimeter). Here, the semi-perimeter is $(10+10+12)/2 = 16$. Thus the area equals $\sqrt{(16)(16-10)(16-10)(16-12)} = \sqrt{16*6*6*4} = 48.$ We know that the width of the rectangle is $4$, so $48/4 = 12$, which is the length. The perimeter of the rectangle is $2(4+12) =$ $\boxed{\textbf{(D)}\ 32}$.

(Solution by Flamedragon)

Solution 3

Note that a triangle with side lengths $10,10,12$ is essentially $2$ “6,8,10” right triangles stuck together. Hence, the height is $8$, and our area is $48$.

So, the length of the rectangle is $\frac{48}{4}=12$, and our perimeter $P=2(4+12)=\boxed{\textbf{(D)}\ 32}$

Video Solution

https://youtu.be/UGLvhMp67Ag

~Education, the Study of Everything

Video Solution=

https://youtu.be/I3yihAO87CE?t=89

~IceMatrix

See Also

2010 AMC 10B (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

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