Difference between revisions of "1975 AHSME Problems/Problem 28"

Latest revision as of 23:49, 11 May 2023

Problem 28

In $\triangle ABC$ shown in the adjoining figure, $M$ is the midpoint of side $BC, AB=12$ and $AC=16$ . Points $E$ and $F$ are taken on $AC$ and $AB$ , respectively, and lines $EF$ and $AM$ intersect at $G$ . If $AE=2AF$ then $\frac{EG}{GF}$ equals

$[asy] draw((0,0)--(12,0)--(14,7.75)--(0,0)); draw((0,0)--(13,3.875)); draw((5,0)--(8.75,4.84)); label("A", (0,0), S); label("B", (12,0), S); label("C", (14,7.75), E); label("E", (8.75,4.84), N); label("F", (5,0), S); label("M", (13,3.875), E); label("G", (7,1)); [/asy]$

$\textbf{(A)}\ \frac{3}{2} \qquad \textbf{(B)}\ \frac{4}{3} \qquad \textbf{(C)}\ \frac{5}{4} \qquad \textbf{(D)}\ \frac{6}{5}\\ \qquad \textbf{(E)}\ \text{not enough information to solve the problem}$

Solution

Here, we use Mass Points. Let $AF = x$ . We then have $AE = 2x$ , $EC = 16-2x$ , and $FB = 12 - x$ Let $B$ have a mass of $2$ . Since $M$ is the midpoint, $C$ also has a mass of $2$ . Looking at segment $AB$ , we have $2 \cdot (12-x) = \text{m}A_{AB} \cdot x$ So $\text{m}A_{AB} = \frac{24-2x}{x}$ Looking at segment $AC$ ,we have $2 \cdot (16-2x) = \text{m}A_{AC} \cdot 2x$ So $\text{m}A_{AC} = \frac{16-2x}{x}$ From this, we get $\text{m}E = \frac{16-2x}{x} + 2 \Rrightarrow \text{m}E = \frac{16}{x}$ and $\text{m}F = \frac{24-2x}{x} + 2 \Rrightarrow \text{m}F = \frac{24}{x}$ We want the value of $\frac{EG}{GF}$ . This can be written as $\frac{EG}{GF} = \frac{\text{m}F}{\text{m}E}$ Thus $\frac{\text{m}F}{\text{m}E} = \frac {\frac{24}{x}}{\frac{16}{x}} = \frac{3}{2}$ $\boxed{A}$

~JustinLee2017

Solution 2

Since we only care about a ratio $EG/GF$ , and since we are given $M$ being the midpoint of $\overline{BC}$ , we realize we can conveniently also choose $E$ to be the midpoint of $\overline{AC}$ . (we're free to choose any point $E$ on $\overline{AC}$ as long as $AE$ is twice $AF$ , the constraint given in the problem). This means $AE = 16/2 = 8$ , and $AF = 4$ . We then connect $ME$ which creates similar triangles $\triangle EMC$ and $\triangle ABC$ by SAS, and thus generates parallel lines $\overline{EM}$ and $\overline{AB}$ . This also immediately gives us similar triangles $\triangle AFG \sim \triangle MEG, => EG/FG = ME/AF = 6/4 = \boxed{3/2}$ (note that $ME = 6$ because $ME/AB$ is in $1:2$ ratio).

~afroromanian

Solution 3

In order to find $\frac{EG}{FG}$ , we can apply the law of sine to this model. Let:

$\angle MAC=\alpha, \; \angle MAB=\beta, \; \angle AGE=\gamma,\; \angle AMC=\delta$

Then, in the $\triangle AMC$ and $\triangle AMB$ :

$\frac{MC}{sin\alpha}=\frac{AC}{sin\delta};$ $\frac{MB}{sin\beta}=\frac{AB}{sin\delta}=\frac{MC}{sin\alpha}\cdot\frac{3}{4};$ $\frac{sin\alpha}{sin\beta}=\frac{3}{4}$

In the $\triangle AGE$ and $\triangle AGF$ :

$\frac{FG}{sin\beta}=\frac{AF}{sin\gamma};$ $\frac{EG}{sin\alpha}=\frac{AE}{sin\gamma};$ $\frac{2FG}{sin\beta}=\frac{EG}{sin\alpha};$ $\frac{EG}{FG}=\frac{2sin\alpha}{sin\beta}=\frac{3}{2}$

Hence, our answer is A.

-VSN

@@ Line 46: / Line 46: @@
 ~JustinLee2017
+==Solution 2==
+Since we only care about a ratio <math>EG/GF</math>, and since we are given <math>M</math> being the midpoint of <math>\overline{BC}</math>, we realize we can conveniently also choose <math>E</math> to be the midpoint of <math>\overline{AC}</math>. (we're free to choose any point <math>E</math> on <math>\overline{AC}</math> as long as <math>AE</math> is twice <math>AF</math>, the constraint given in the problem). This means <math>AE = 16/2 = 8</math>, and <math>AF = 4</math>. We then connect <math>ME</math> which creates similar triangles <math>\triangle EMC</math> and <math>\triangle ABC</math> by SAS, and thus generates parallel lines <math>\overline{EM}</math> and <math>\overline{AB}</math>. This also immediately gives us similar triangles <math>\triangle AFG \sim \triangle MEG, => EG/FG = ME/AF = 6/4 = \boxed{3/2}</math> (note that <math>ME = 6</math> because <math>ME/AB</math> is in <math>1:2</math> ratio).
+~afroromanian
+==Solution 3==
+In order to find <math>\frac{EG}{FG}</math>, we can apply the law of sine to this model. Let:
+<math>\angle MAC=\alpha, \; \angle MAB=\beta, \; \angle AGE=\gamma,\; \angle AMC=\delta</math>
+Then, in the <math>\triangle AMC</math> and <math>\triangle AMB</math>:
+<math>\frac{MC}{sin\alpha}=\frac{AC}{sin\delta};</math>
+<math>\frac{MB}{sin\beta}=\frac{AB}{sin\delta}=\frac{MC}{sin\alpha}\cdot\frac{3}{4};</math>
+<math>\frac{sin\alpha}{sin\beta}=\frac{3}{4}</math>
+In the <math>\triangle AGE</math> and <math>\triangle AGF</math>:
+<math>\frac{FG}{sin\beta}=\frac{AF}{sin\gamma};</math>
+<math>\frac{EG}{sin\alpha}=\frac{AE}{sin\gamma};</math>
+<math>\frac{2FG}{sin\beta}=\frac{EG}{sin\alpha};</math>
+<math>\frac{EG}{FG}=\frac{2sin\alpha}{sin\beta}=\frac{3}{2}</math>
+Hence, our answer is A.
+-VSN
 ==See Also==
 {{AHSME box|year=1975|num-b=27|num-a=29}}
 {{MAA Notice}}

1975 AHSME (Problems • Answer Key • Resources)
Preceded by Problem 27	Followed by Problem 29
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