Difference between revisions of "1962 AHSME Problems/Problem 23"
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<math>\textbf{(C)}\ \text{determined only if B is an acute angle} \qquad</math> | <math>\textbf{(C)}\ \text{determined only if B is an acute angle} \qquad</math> | ||
− | <math>\textbf{(D)}\ \text{determined only | + | <math>\textbf{(D)}\ \text{determined only if ABC is an acute triangle} \qquad</math> |
<math>\textbf{(E)}\ \text{none of these is correct} </math> | <math>\textbf{(E)}\ \text{none of these is correct} </math> |
Revision as of 21:39, 10 April 2023
Problem
In triangle ,
is the altitude to
and
is the altitude to
. If the lengths of
,
, and
are known, the length of
is:
Solution
We can actually determine the length of no matter what type of angles
and
are. This can be easily proved through considering all possible cases, though for the purposes of this solution, we'll show that we can determine
if
is an obtuse angle.
Let's see what happens when is an obtuse angle.
by SSS, so
. Hence
. Since we've determined the length of
even though we have an obtuse angle,
is not
determined by what type of angle
may be. Hence our answer is
.