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Difference between revisions of "1950 AHSME Problems/Problem 49"
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==Solution==
 
==Solution==
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The locus of the median's endpoint on <math>BC</math> is the circle about <math>A</math> and of radius <math>1\frac{1}{2}</math> inches.  The locus of the vertex <math>C</math> is then the circle twice as big and twice as far from <math>B</math>, i.e. of radius <math>3</math> inches and with center <math>4</math> inches from <math>B</math> along <math>BA</math> : <math>\textbf{(D)}</math>.
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The locus of the median's endpoint on <math>BC</math> is the circle about <math>A</math> and of radius <math>1\frac{1}{2}</math> inches.  The locus of the vertex <math>C</math> is then the circle twice as big and twice as far from <math>B</math>, i.e. of radius <math>3</math> inches and with center <math>4</math> inches from <math>B</math> along <math>BA</math> which means that our answer is: <math>\textbf{(D)}</math>.
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==See Also==
 
==See Also==
 
{{AHSME 50p box|year=1950|num-b=48|num-a=50}}
 
{{AHSME 50p box|year=1950|num-b=48|num-a=50}}

Revision as of 21:09, 22 February 2017

Problem

A triangle has a fixed base $AB$ that is $2$ inches long. The median from $A$ to side $BC$ is $1\frac{1}{2}$ inches long and can have any position emanating from $A$. The locus of the vertex $C$ of the triangle is:

$\textbf{(A)}\ \text{A straight line }AB,1\dfrac{1}{2}\text{ inches from }A \qquad\\ \textbf{(B)}\ \text{A circle with }A\text{ as center and radius }2\text{ inches} \qquad\\ \textbf{(C)}\ \text{A circle with }A\text{ as center and radius }3\text{ inches} \qquad\\ \textbf{(D)}\ \text{A circle with radius }3\text{ inches and center }4\text{ inches from }B\text{ along } BA \qquad\\ \textbf{(E)}\ \text{An ellipse with }A\text{ as focus}$

Solution

The locus of the median's endpoint on $BC$ is the circle about $A$ and of radius $1\frac{1}{2}$ inches. The locus of the vertex $C$ is then the circle twice as big and twice as far from $B$, i.e. of radius $3$ inches and with center $4$ inches from $B$ along $BA$ which means that our answer is: $\textbf{(D)}$.

See Also

1950 AHSC (Problems • Answer Key • Resources)
Preceded by
Problem 48 		Followed by
Problem 50
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All AHSME Problems and Solutions

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