Difference between revisions of "2007 AMC 12B Problems/Problem 23"

(Solution 3)
(Solution 3)
(3 intermediate revisions by one other user not shown)
Line 58: Line 58:
 
== Solution 3 ==
 
== Solution 3 ==
  
Let the right triable be <math>\triangle ABC</math>, the two legs be <math>a</math> and <math>b</math>, the hypotenuse be <math>c</math>
+
Let <math>a</math> and <math>b</math> be the two legs of the triangle, and <math>c</math> be the hypotenuse.
  
By using <math>[ABC] = r\cdot s</math> we get:
+
By using <math>Area = \frac{r}{2} (a+b+c)</math>, where <math>r</math> is the in-radius, we get:
  
<cmath>3(a+b+c) = r \cdot \frac{a+b+c}{2}</cmath>
+
<cmath>3(a+b+c) = \frac{r}{2} (a+b+c)</cmath>
 
<cmath>r=6</cmath>
 
<cmath>r=6</cmath>
<cmath>r = \frac{a+b-c}{2}</cmath>
+
 
 +
In right triangle, <math>r = \frac{a+b-c}{2}</math>
 
<cmath>a+b-c = 12</cmath>
 
<cmath>a+b-c = 12</cmath>
 
<cmath>c = a + b - 12</cmath>
 
<cmath>c = a + b - 12</cmath>
Line 80: Line 81:
 
<cmath>(a - 12)(b - 12) = 72</cmath>
 
<cmath>(a - 12)(b - 12) = 72</cmath>
  
As <math>72 = 2^3 \cdot 3^2</math>, there are <math>\frac{(3+1)(2+1)}{2} = 6</math> solutions, <math>\boxed{\textbf{(A) } 6}</math>
+
As <math>72 = 2^3 \cdot 3^2</math>, there are <math>\frac{(3+1)(2+1)}{2} = 6</math> solutions, <math>\boxed{\textbf{(A) } 6}</math>.
  
 
~[https://artofproblemsolving.com/wiki/index.php/User:Isabelchen isabelchen]
 
~[https://artofproblemsolving.com/wiki/index.php/User:Isabelchen isabelchen]
 +
 +
== Solution 4 ==
 +
All pythagorean triples can be parametrized in the form <math>(a, b, c) = k(r^2 - s^2), k(2rs), k(r^2 + s^2)</math> for positive integers <math>k, r, s</math>. The area being triple the perimeter implies that <cmath>k^2(r^2 - s^2)rs = 3(k(r^2 - s^2) + k(2rs) + k(r^2 + s^2)).</cmath> This can be simplified to get <cmath>ks(r - s) = 6.</cmath> Now, we get the triples <cmath>(k, r, s) = (1, 7, 1), (1, 5, 2), (1, 5, 3), (1, 7, 6), (2, 4, 1), (2, 4, 3), (3, 3, 1), (3, 3, 2), (6, 2, 1).</cmath> However, the ones where <math>r</math> and <math>s</math> are not different signs and relatively prime are redundant, so we get <math>6</math> triples total.
  
 
==See Also==
 
==See Also==
 
{{AMC12 box|year=2007|ab=B|num-b=22|num-a=24}}
 
{{AMC12 box|year=2007|ab=B|num-b=22|num-a=24}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 20:00, 22 May 2023

Problem

How many non-congruent right triangles with positive integer leg lengths have areas that are numerically equal to $3$ times their perimeters?

$\mathrm {(A)} 6\qquad \mathrm {(B)} 7\qquad \mathrm {(C)} 8\qquad \mathrm {(D)} 10\qquad \mathrm {(E)} 12$

Solution 1

Let $a$ and $b$ be the two legs of the triangle.

We have $\frac{1}{2}ab = 3(a+b+c)$.

Then $ab=6 \left(a+b+\sqrt {a^2 + b^2}\right)$.

We can complete the square under the root, and we get, $ab=6 \left(a+b+\sqrt {(a+b)^2 - 2ab}\right)$.

Let $ab=p$ and $a+b=s$, we have $p=6 \left(s+ \sqrt {s^2 - 2p}\right)$.

After rearranging, squaring both sides, and simplifying, we have $p=12s-72$.


Putting back $a$ and $b$, and after factoring using Simon's Favorite Factoring Trick, we've got $(a-12)(b-12)=72$.


Factoring 72, we get 6 pairs of $a$ and $b$


$(13, 84), (14, 48), (15, 36), (16, 30), (18, 24), (20, 21).$


And this gives us $6$ solutions $\Rightarrow \mathrm{(A)}$.


Alternatively, note that $72 = 2^3 \cdot 3^2$. Then 72 has $(3+1)(2+1) = (4)(3) = 12$ factors. However, half of these are repeats, so we have $\frac{12}{2} = 6$ solutions.

Solution 2

We will proceed by using the fact that $[ABC] = r\cdot s$, where $r$ is the radius of the incircle and $s$ is the semiperimeter $\left(s = \frac{p}{2}\right)$.

We are given $[ABC] = 3p = 6s \Rightarrow rs = 6s \Rightarrow r = 6$.

The incircle of $ABC$ breaks the triangle's sides into segments such that $AB = x + y$, $BC = x + z$ and $AC = y + z$. Since ABC is a right triangle, one of $x$, $y$ and $z$ is equal to its radius, 6. Let's assume $z = 6$.

The side lengths then become $AB = x + y$, $BC = x + 6$ and $AC = y + 6$. Plugging into Pythagorean's theorem:

$(x + y)^2 = (x+6)^2 + (y + 6)^2$

$x^2 + 2xy + y^2 = x^2 + 12x + 36 + y^2 + 12y + 36$

$2xy - 12x - 12y = 72$

$xy - 6x - 6y = 36$

$(x - 6)(y - 6) - 36 = 36$

$(x - 6)(y - 6) = 72$

We can factor $72$ to arrive with $6$ pairs of solutions: $(7, 78), (8,42), (9, 30), (10, 24), (12, 18),$ and $(14, 15) \Rightarrow \mathrm{(A)}$.

Solution 3

Let $a$ and $b$ be the two legs of the triangle, and $c$ be the hypotenuse.

By using $Area = \frac{r}{2} (a+b+c)$, where $r$ is the in-radius, we get:

\[3(a+b+c) = \frac{r}{2} (a+b+c)\] \[r=6\]

In right triangle, $r = \frac{a+b-c}{2}$ \[a+b-c = 12\] \[c = a + b - 12\]


By the triangle's area we get:

\[\frac{ab}{2} = 6 \cdot \frac{a+b+c}{2}\] \[ab = 6(a+b+c)\]

By substituting $c$ in:

\[ab = 6(a+b+a + b - 12)\] \[ab - 12a - 12b + 72 = 0\] \[(a - 12)(b - 12) = 72\]

As $72 = 2^3 \cdot 3^2$, there are $\frac{(3+1)(2+1)}{2} = 6$ solutions, $\boxed{\textbf{(A) } 6}$.

~isabelchen

Solution 4

All pythagorean triples can be parametrized in the form $(a, b, c) = k(r^2 - s^2), k(2rs), k(r^2 + s^2)$ for positive integers $k, r, s$. The area being triple the perimeter implies that \[k^2(r^2 - s^2)rs = 3(k(r^2 - s^2) + k(2rs) + k(r^2 + s^2)).\] This can be simplified to get \[ks(r - s) = 6.\] Now, we get the triples \[(k, r, s) = (1, 7, 1), (1, 5, 2), (1, 5, 3), (1, 7, 6), (2, 4, 1), (2, 4, 3), (3, 3, 1), (3, 3, 2), (6, 2, 1).\] However, the ones where $r$ and $s$ are not different signs and relatively prime are redundant, so we get $6$ triples total.

See Also

2007 AMC 12B (ProblemsAnswer KeyResources)
Preceded by
Problem 22
Followed by
Problem 24
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. AMC logo.png