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Difference between revisions of "1982 AHSME Problems/Problem 23"

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This solution uses the same variable definitions as Solution 1 does. Moreover, we conclude that <math>\cos\theta=\frac{n+5}{2(n+2)}</math> from the second paragraph of Solution 1.
 
This solution uses the same variable definitions as Solution 1 does. Moreover, we conclude that <math>\cos\theta=\frac{n+5}{2(n+2)}</math> from the second paragraph of Solution 1.
  
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Let the brackets denote areas. We write <math>[ABC]</math> using <math>\sin\angle A</math> and Heron's Formula, respectively:
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Let the brackets denote areas. We write <math>[ABC]</math> using <math>\sin\angle A</math> and Heron's Formula, respectively: <cmath>[ABC]=\frac12 bc\sin\theta=\sqrt{s(s-a)(s-b)(s-c)},</cmath> where <math>s=\frac{a+b+c}{2}</math> is the semiperimeter of <math>\triangle ABC.</math>
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 +
Equating the last two expressions and then simplifying, we have <cmath>\sin\theta=\frac{2\sqrt{s(s-a)(s-b)(s-c)}}{bc}=\frac{(n+1)\sqrt{3(n+3)(n-1)}}{2(n+2)}.</cmath>
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Recall that <math>\sin^2\theta+\cos^2\theta=1</math> holds for all <math>\theta.</math>
  
 
~MRENTHUSIASM
 
~MRENTHUSIASM

Revision as of 02:10, 15 September 2021

Problem

The lengths of the sides of a triangle are consecutive integers, and the largest angle is twice the smallest angle. The cosine of the smallest angle is

$\textbf{(A)}\ \frac{3}{4}\qquad \textbf{(B)}\ \frac{7}{10}\qquad \textbf{(C)}\ \frac{2}{3}\qquad \textbf{(D)}\ \frac{9}{14}\qquad \textbf{(E)}\ \text{none of these}$

Solution 1

In $\triangle ABC,$ let $a=n,b=n+1,c=n+2,$ and $\angle A=\theta$ for some positive integer $n.$ We are given that $\angle C=2\theta,$ and we need $\cos\theta.$

We apply the Law of Cosines to solve for $\cos\angle A:$ \[\cos\theta=\frac{b^2+c^2-a^2}{2bc}=\frac{n+5}{2(n+2)}.\]

Let the brackets denote areas. We write $[ABC]$ in terms of $\sin\angle A$ and $\sin\angle C,$ respectively: \[[ABC]=\frac12 bc\sin\theta=\frac12 ab\sin(2\theta).\] Recall that $\sin(2\theta)=2\sin\theta\cos\theta$ holds for all $\theta.$ Equating the last two expressions and then simplifying, we have \[\cos\theta=\frac{c}{2a}=\frac{n+2}{2n}.\] Equating the expressions for $\cos\theta,$ we get \[\frac{n+5}{2(n+2)}=\frac{n+2}{2n},\] from which $n=4.$ By substitution, the answer is $\cos\theta=\boxed{\textbf{(A)}\ \frac{3}{4}}.$

~MRENTHUSIASM

Solution 2

This solution uses the same variable definitions as Solution 1 does. Moreover, we conclude that $\cos\theta=\frac{n+5}{2(n+2)}$ from the second paragraph of Solution 1.

Let the brackets denote areas. We write $[ABC]$ using $\sin\angle A$ and Heron's Formula, respectively: \[[ABC]=\frac12 bc\sin\theta=\sqrt{s(s-a)(s-b)(s-c)},\] where $s=\frac{a+b+c}{2}$ is the semiperimeter of $\triangle ABC.$

Equating the last two expressions and then simplifying, we have \[\sin\theta=\frac{2\sqrt{s(s-a)(s-b)(s-c)}}{bc}=\frac{(n+1)\sqrt{3(n+3)(n-1)}}{2(n+2)}.\] Recall that $\sin^2\theta+\cos^2\theta=1$ holds for all $\theta.$

~MRENTHUSIASM

Solution 3

This solution uses the same variable definitions as Solution 1 does. Moreover, we conclude that $\cos\theta=\frac{n+5}{2(n+2)}$ from the second paragraph of Solution 1.

We apply the Law of Cosines to solve for $\cos\angle C:$ \[\cos(2\theta)=\frac{a^2+b^2-c^2}{2ab}=\frac{n-3}{2n}.\] By the Double-Angle Formula $\cos(2\theta)=2\cos^2\theta-1,$ we have \[2\left(\frac{n+5}{2(n+2)}\right)^2-1=\frac{n-3}{2n},\] from which $n=-3,-\frac12,4.$ Recall that $n$ is a positive integer, so $n=4.$ By substitution, the answer is $\cos\theta=\boxed{\textbf{(A)}\ \frac{3}{4}}.$

~MRENTHUSIASM

See Also

1982 AHSME (Problems • Answer Key • Resources)
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 • 26 • 27 • 28 • 29 • 30
All AHSME Problems and Solutions

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