1957 AHSME Problems/Problem 1

The number of distinct lines representing the altitudes, medians, and interior angle bisectors of a triangle that is isosceles, but not equilateral, is:

$\textbf{(A)}\ 9\qquad \textbf{(B)}\ 7\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 5\qquad \textbf{(E)}\ 3$

Solution

[asy] size(2cm); draw((-3,0)--(0,4)--(3,0)--cycle); draw((0,0)--(0,4), red);  draw((-3,0)--(0.84, 2.88), green); draw((-3,0)--(1.5, 2), green); draw((-3,0)--(1.636, 1.818), green);  draw((3,0)--(-0.84, 2.88), blue); draw((3,0)--(-1.5, 2), blue); draw((3,0)--(-1.636, 1.818), blue); [/asy]

As shown in the diagram above, all nine altitudes, medians, and interior angle bisectors are distinct, except for the three coinciding lines from the vertex opposite to the base. Thusly, there are $7$ distinct lines, so our answer is $\boxed{\textbf{(B)}}$, and we are done.

See also

1957 AHSC (ProblemsAnswer KeyResources)
Preceded by
First Problem
Followed by
Problem 2
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 41 42 43 44 45 46 47 48 49 50
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