1962 AHSME Problems/Problem 15

Revision as of 15:10, 29 January 2018 by Mitko pitko (talk | contribs) (Solution)

Problem

Given triangle $ABC$ with base $AB$ fixed in length and position. As the vertex $C$ moves on a straight line, the intersection point of the three medians moves on:

$\textbf{(A)}\ \text{a circle}\qquad\textbf{(B)}\ \text{a parabola}\qquad\textbf{(C)}\ \text{an ellipse}\qquad\textbf{(D)}\ \text{a straight line}\qquad\textbf{(E)}\ \text{a curve here not listed}$

Solution

This problem needs a solution. If you have a solution for it, please help us out by adding it.

Let $CM$ be the median through vertex $C$, and let $G$ be the point of intersection of the triangle's medians.

Let $CH$ be the altitude of the triangle through vertex $C$ and $GP$ be the distance from $G$ to $AB$, with the point $P$ laying on $AB$.

Using Thales' intercept theorem, we derive the proportion:


$\frac{GP}{CH} = \frac{GM}{CM}$


The fraction $\frac{GM}{CM}$ in any triangle is

equal to $\frac{1}{3}$ . Therefore $GP = \frac{CH}{3}$ .


Since the problem states that the vertex $C$ is moving in a straight line, the length of $CH$ is a constant value. That means that the length of $GP$ is also a constant. Therefore the point $G$ is moving in a straight line.

Answer: D

See Also

1962 AHSC (ProblemsAnswer KeyResources)
Preceded by
Problem 14
Followed by
Problem 16
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 31 32 33 34 35 36 37 38 39 40
All AHSME Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png