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Difference between revisions of "1950 AHSME Problems/Problem 49"

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==Solution==
 
==Solution==
 
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>.
 
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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Let <math>A(a,y_A)</math>, <math>B(b,y_B)</math> and <math>C(x,y)</math>.
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Hence, <math>D\left(\frac{x+b}{2},\frac{y+y_B}{2}\right)</math> is a midpoint of <math>BC</math>.
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Thus, the equation of needed locus is
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<cmath>\left(\frac{x+b}{2}-a\right)^2+\left(\frac{y+y_B}{2}-y_A\right)^2=\left(\frac{3}{2}\right)^2,</cmath>
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which is equation of the circle:
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<cmath>(x-(2a-b))^2+(y-(2y_A-y_B))^2=3^2.</cmath>
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Thus, D) is valid because <cmath>\sqrt{(2a-b-b)^2+(2y_A-y_B-y_B)^2}=2\sqrt{(a-b)^2+(y_A-y_b)^2}=2AB=4.</cmath>
  
 
==See Also==
 
==See Also==

Revision as of 22:08, 9 August 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: $\textbf{(D)}$.

Let $A(a,y_A)$, $B(b,y_B)$ and $C(x,y)$.

Hence, $D\left(\frac{x+b}{2},\frac{y+y_B}{2}\right)$ is a midpoint of $BC$.

Thus, the equation of needed locus is \[\left(\frac{x+b}{2}-a\right)^2+\left(\frac{y+y_B}{2}-y_A\right)^2=\left(\frac{3}{2}\right)^2,\] which is equation of the circle: \[(x-(2a-b))^2+(y-(2y_A-y_B))^2=3^2.\]

Thus, D) is valid because \[\sqrt{(2a-b-b)^2+(2y_A-y_B-y_B)^2}=2\sqrt{(a-b)^2+(y_A-y_b)^2}=2AB=4.\]

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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