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Difference between revisions of "2015 AMC 12A Problems/Problem 20"

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<math>\textbf{(A) }3\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad\textbf{(E) }8</math>
 
<math>\textbf{(A) }3\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad\textbf{(E) }8</math>
  
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==Solution==
+
==Solution 1 ==
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===Solution 1 ===
 
 
The area of <math>T</math> is <math>\dfrac{1}{2} \cdot 8 \cdot 3 = 12</math> and the perimeter is 18.
 
The area of <math>T</math> is <math>\dfrac{1}{2} \cdot 8 \cdot 3 = 12</math> and the perimeter is 18.
  
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Thus <math>12 = \dfrac{1}{2} b \sqrt{a^2 - (\dfrac{b}{2})^2}</math>, so <math>48 = b \sqrt{4a^2 - b^2} = b \sqrt{(18 - b)^2 - b^2} = b \sqrt{324 - 36b}</math>.
 
Thus <math>12 = \dfrac{1}{2} b \sqrt{a^2 - (\dfrac{b}{2})^2}</math>, so <math>48 = b \sqrt{4a^2 - b^2} = b \sqrt{(18 - b)^2 - b^2} = b \sqrt{324 - 36b}</math>.
  
−
We square and divide 36 from both sides to obtain <math>64 = b^2 (9 - b)</math>, so <math>b^3 - 9b^2 + 64 = 0</math>. Since we know <math>b = 8</math> is a solution, we divide by <math>b - 8</math> to get the other solution. Thus, we have <math>b^2 - b - 8 = 0</math>, so <math>b = \dfrac{1 + \sqrt{33}}{2} < \dfrac{1 + 6}{2} = 3.5.</math> The answer is <math>\textbf{(A)}</math>.
+
We square and divide 36 from both sides to obtain <math>64 = b^2 (9 - b)</math>, so <math>b^3 - 9b^2 + 64 = 0</math>. Since we know <math>b = 8</math> is a solution, we divide by <math>b - 8</math> to get the other solution. Thus, <math>b^2 - b - 8 = 0</math>, so <math>b = \dfrac{1 + \sqrt{33}}{2} < \dfrac{1 + 6}{2} = 3.5.</math> The answer is <math>\boxed{\textbf{(A) }3}</math>.
  
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===Solution 2===
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===Solution 1.1===
 +
The area is <math>12</math>, the semiperimeter is <math>9</math>, and <math>a = 9 - \frac12b</math>. Using Heron's formula, <math>\sqrt{9\left(\frac{b}{2}\right)\left(\frac{b}{2}\right)(9-b)} = 12</math>. Squaring both sides and simplifying, we have <math>-b^3+9b^2-64=0</math>. Since we know <math>b = 8</math> is a solution, we divide by <math>b - 8</math> to get the other solution. Thus, <math>b^2 - b - 8 = 0</math>, so <math>b = \dfrac{1 + \sqrt{33}}{2} < \dfrac{1 + 6}{2} = 3.5.</math> The answer is <math>\boxed{\textbf{(A) }3}</math>.
 +
 
 +
==Solution 2==
  
 
Triangle <math>T</math>, being isosceles, has an area of <math>\frac{1}{2}(8)\sqrt{5^2-4^2}=12</math> and a perimeter of <math>5+5+8=18</math>.
 
Triangle <math>T</math>, being isosceles, has an area of <math>\frac{1}{2}(8)\sqrt{5^2-4^2}=12</math> and a perimeter of <math>5+5+8=18</math>.
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Utilizing the quadratic formula gives
 
Utilizing the quadratic formula gives
 
<cmath>b=\frac{1\pm\sqrt{33}}{2}</cmath>
 
<cmath>b=\frac{1\pm\sqrt{33}}{2}</cmath>
−
We clearly must pick the positive solution. Note that <math>5<\sqrt{33}<6</math>, and so <math>{3<\frac{1+\sqrt{33}}{2}<\frac{7}{2}}</math>, which clearly gives an answer of <math>\fbox{A}</math>, as desired.
+
We clearly must pick the positive solution. Note that <math>5<\sqrt{33}<6</math>, and so <math>{3<\frac{1+\sqrt{33}}{2}<\frac{7}{2}}</math>, which clearly gives an answer of <math>\boxed{\textbf{(A) }3}</math>, as desired.
  
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===Solution 3===
+
==Solution 3==
 
Triangle T has perimeter <math>5 + 5 + 8 = 18</math> so <math>18 = 2a + b</math>.
 
Triangle T has perimeter <math>5 + 5 + 8 = 18</math> so <math>18 = 2a + b</math>.
  
−
Using Heron's, we get <math>\sqrt{(9)(4)^2(1)} = \sqrt{(\frac{2a+b}{2})(\frac{b}{2})^2(\frac{2a-b}{2})}</math>.
+
Using Heron's, we get <math>\sqrt{(9)(4)^2(1)} = \sqrt{(\frac{2a+b}{2})\left(\frac{b}{2}\right)^2(\frac{2a-b}{2})}</math>.
  
 
We know that <math>2a + b = 18</math> from above so we plug that in, and we also know that then <math>2a - b = 18 - 2b</math>.
 
We know that <math>2a + b = 18</math> from above so we plug that in, and we also know that then <math>2a - b = 18 - 2b</math>.
  
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<math>12 = 3\frac{b}{2}\sqrt{9-b}</math>
+
<math>12 = \frac{3b}{2}\sqrt{9-b}</math>
  
 
<math>64 = 9b^2 - b^3</math>
 
<math>64 = 9b^2 - b^3</math>
  
−
We plug in 3 for <math>b</math> in the LHS, and we get 54 which is too low. We plug in 4 for <math>b</math> in the LHS, and we get 80 which is too high. We now know that b is some number between 3 and 4.  
+
We plug in 3 for <math>b</math> in the LHS, and we get 54 which is too low. We plug in 4 for <math>b</math> in the LHS, and we get 80 which is too high. We now know that <math>b</math> is some number between 3 and 4.  
  
−
If <math>b \geq 3.5</math>, then we would round up to 4, but if <math>b < 3.5</math>, then we would round down to 3. So let us plug in 3.5 for b.  
+
If <math>b \geq 3.5</math>, then we would round up to 4, but if <math>b < 3.5</math>, then we would round down to 3. So let us plug in 3.5 for <math>b</math>.  
  
−
We get 67.375 which is too high, so we know that <math>b < 3.5</math>.
+
We get <math>67.375</math> which is too high, so we know that <math>b < 3.5</math>.
  
−
The answer is <math>3</math>. <math>\textbf{(A)}</math>
+
Thus the answer is <math>\boxed{\textbf{(A) }3}</math>
  
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===Operation Descartes===
+
==Solution 4 (Operation Descartes)==
 
For this new triangle, say its legs have length <math>d</math> and the base length <math>2c</math>. To see why I did this, draw the triangle on a Cartesian plane where the altitude is part of the y-axis! Then, we notice that <math>c+d=9</math> and <math>c*\sqrt{d^2-c^2}=12</math>. It's better to let a side be some variable so we avoid having to add non-square roots and square-roots!!
 
For this new triangle, say its legs have length <math>d</math> and the base length <math>2c</math>. To see why I did this, draw the triangle on a Cartesian plane where the altitude is part of the y-axis! Then, we notice that <math>c+d=9</math> and <math>c*\sqrt{d^2-c^2}=12</math>. It's better to let a side be some variable so we avoid having to add non-square roots and square-roots!!
  
−
Now, modify the square-root equation with <math>d=9-c</math>; you get <math>c^2*(81-18c)=144</math>, so <math>-18c^3+81c^2=144</math>. Divide by <math>-9</math> to get <math>2c^3-9c^2+16=0</math>. Obviously, <math>c=4</math> is a root as established by triangle <math>T</math>! So, use synthetic division to obtain <math>2c^2-c-4=0</math>, upon which <math>c=\frac{1+\sqrt{33}}{4}</math>, which is closest to <math>\frac{3}{2}</math> (as opposed to <math>2</math>). That's enough to confirm that the answer has to be <math>\textbf{A}</math>.
+
Now, modify the square-root equation with <math>d=9-c</math>; you get <math>c^2*(81-18c)=144</math>, so <math>-18c^3+81c^2=144</math>. Divide by <math>-9</math> to get <math>2c^3-9c^2+16=0</math>. Obviously, <math>c=4</math> is a root as established by triangle <math>T</math>! So, use synthetic division to obtain <math>2c^2-c-4=0</math>, upon which <math>c=\frac{1+\sqrt{33}}{4}</math>, which is closest to <math>\frac{3}{2}</math> (as opposed to <math>2</math>). That's enough to confirm that the answer has to be <math>\boxed{\textbf{(A) }3}</math>.
 +
 
 +
==Solution 5 (When You're Running Out of Time)==
 +
Since triangles <math>T'</math> and <math>T</math> have the same area and the same perimeter,
 +
<math>2a+b=18</math> and <math>9*(9-a)^2(9-b) = 9*4^2*1</math>
 +
By trying each answer choice, it is clear that the answer is <math>\boxed{\textbf{(A) }3}</math>.
  
 
== See Also ==
 
== See Also ==
 
{{AMC12 box|year=2015|ab=A|num-b=19|num-a=21}}
 
{{AMC12 box|year=2015|ab=A|num-b=19|num-a=21}}

Latest revision as of 21:32, 7 June 2021

Problem

Isosceles triangles $T$ and $T'$ are not congruent but have the same area and the same perimeter. The sides of $T$ have lengths $5$, $5$, and $8$, while those of $T'$ have lengths $a$, $a$, and $b$. Which of the following numbers is closest to $b$?

$\textbf{(A) }3\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad\textbf{(E) }8$

Solution 1

The area of $T$ is $\dfrac{1}{2} \cdot 8 \cdot 3 = 12$ and the perimeter is 18.

The area of $T'$ is $\dfrac{1}{2} b \sqrt{a^2 - (\dfrac{b}{2})^2}$ and the perimeter is $2a + b$.

Thus $2a + b = 18$, so $2a = 18 - b$.

Thus $12 = \dfrac{1}{2} b \sqrt{a^2 - (\dfrac{b}{2})^2}$, so $48 = b \sqrt{4a^2 - b^2} = b \sqrt{(18 - b)^2 - b^2} = b \sqrt{324 - 36b}$.

We square and divide 36 from both sides to obtain $64 = b^2 (9 - b)$, so $b^3 - 9b^2 + 64 = 0$. Since we know $b = 8$ is a solution, we divide by $b - 8$ to get the other solution. Thus, $b^2 - b - 8 = 0$, so $b = \dfrac{1 + \sqrt{33}}{2} < \dfrac{1 + 6}{2} = 3.5.$ The answer is $\boxed{\textbf{(A) }3}$.

Solution 1.1

The area is $12$, the semiperimeter is $9$, and $a = 9 - \frac12b$. Using Heron's formula, $\sqrt{9\left(\frac{b}{2}\right)\left(\frac{b}{2}\right)(9-b)} = 12$. Squaring both sides and simplifying, we have $-b^3+9b^2-64=0$. Since we know $b = 8$ is a solution, we divide by $b - 8$ to get the other solution. Thus, $b^2 - b - 8 = 0$, so $b = \dfrac{1 + \sqrt{33}}{2} < \dfrac{1 + 6}{2} = 3.5.$ The answer is $\boxed{\textbf{(A) }3}$.

Solution 2

Triangle $T$, being isosceles, has an area of $\frac{1}{2}(8)\sqrt{5^2-4^2}=12$ and a perimeter of $5+5+8=18$. Triangle $T'$ similarly has an area of $\frac{1}{2}(b)\bigg(\sqrt{a^2-\frac{b^2}{4}}\bigg)=12$ and $2a+b=18$.

Now we apply our computational fortitude.

\[\frac{1}{2}(b)\bigg(\sqrt{a^2-\frac{b^2}{4}}\bigg)=12\] \[(b)\bigg(\sqrt{a^2-\frac{b^2}{4}}\bigg)=24\] \[(b)\sqrt{4a^2-b^2}=48\] \[b^2(4a^2-b^2)=48^2\] \[b^2(2a+b)(2a-b)=48^2\] Plug in $2a+b=18$ to obtain \[18b^2(2a-b)=48^2\] \[b^2(2a-b)=128\] Plug in $2a=18-b$ to obtain \[b^2(18-2b)=128\] \[2b^3-18b^2+128=0\] \[b^3-9b^2+64=0\] We know that $b=8$ is a valid solution by $T$. Factoring out $b-8$, we obtain \[(b-8)(b^2-b-8)=0 \Rightarrow b^2-b-8=0\] Utilizing the quadratic formula gives \[b=\frac{1\pm\sqrt{33}}{2}\] We clearly must pick the positive solution. Note that $5<\sqrt{33}<6$, and so ${3<\frac{1+\sqrt{33}}{2}<\frac{7}{2}}$, which clearly gives an answer of $\boxed{\textbf{(A) }3}$, as desired.

Solution 3

Triangle T has perimeter $5 + 5 + 8 = 18$ so $18 = 2a + b$.

Using Heron's, we get $\sqrt{(9)(4)^2(1)} = \sqrt{(\frac{2a+b}{2})\left(\frac{b}{2}\right)^2(\frac{2a-b}{2})}$.

We know that $2a + b = 18$ from above so we plug that in, and we also know that then $2a - b = 18 - 2b$.

$12 = \frac{3b}{2}\sqrt{9-b}$

$64 = 9b^2 - b^3$

We plug in 3 for $b$ in the LHS, and we get 54 which is too low. We plug in 4 for $b$ in the LHS, and we get 80 which is too high. We now know that $b$ is some number between 3 and 4.

If $b \geq 3.5$, then we would round up to 4, but if $b < 3.5$, then we would round down to 3. So let us plug in 3.5 for $b$.

We get $67.375$ which is too high, so we know that $b < 3.5$.

Thus the answer is $\boxed{\textbf{(A) }3}$

Solution 4 (Operation Descartes)

For this new triangle, say its legs have length $d$ and the base length $2c$. To see why I did this, draw the triangle on a Cartesian plane where the altitude is part of the y-axis! Then, we notice that $c+d=9$ and $c*\sqrt{d^2-c^2}=12$. It's better to let a side be some variable so we avoid having to add non-square roots and square-roots!!

Now, modify the square-root equation with $d=9-c$; you get $c^2*(81-18c)=144$, so $-18c^3+81c^2=144$. Divide by $-9$ to get $2c^3-9c^2+16=0$. Obviously, $c=4$ is a root as established by triangle $T$! So, use synthetic division to obtain $2c^2-c-4=0$, upon which $c=\frac{1+\sqrt{33}}{4}$, which is closest to $\frac{3}{2}$ (as opposed to $2$). That's enough to confirm that the answer has to be $\boxed{\textbf{(A) }3}$.

Solution 5 (When You're Running Out of Time)

Since triangles $T'$ and $T$ have the same area and the same perimeter, $2a+b=18$ and $9*(9-a)^2(9-b) = 9*4^2*1$ By trying each answer choice, it is clear that the answer is $\boxed{\textbf{(A) }3}$.

See Also

2015 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 19 		Followed by
Problem 21
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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