Difference between revisions of "1962 AHSME Problems/Problem 23"
Robotman1717 (talk | contribs) m (→Problem) |
Erringbubble (talk | contribs) m (SSS is incorrect. We are not given any ratios. the correct one is AA, using the right angle and <B.) |
||
Line 17: | Line 17: | ||
We can actually determine the length of <math>DB</math> no matter what type of angles <math>A</math> and <math>B</math> 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 <math>DB</math> if <math>A</math> is an obtuse angle. | We can actually determine the length of <math>DB</math> no matter what type of angles <math>A</math> and <math>B</math> 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 <math>DB</math> if <math>A</math> is an obtuse angle. | ||
− | Let's see what happens when <math>A</math> is an obtuse angle. <math>\triangle AEB\sim \triangle CDB</math> by | + | Let's see what happens when <math>A</math> is an obtuse angle. <math>\triangle AEB\sim \triangle CDB</math> by AA, so <math>\frac{AE}{CD}=\frac{AB}{DB}</math>. Hence <math>DB=\frac{AE}{CD\times AB}</math>. Since we've determined the length of <math>DB</math> even though we have an obtuse angle, <math>DB</math> is not <math>\bf{only}</math> determined by what type of angle <math>A</math> may be. Hence our answer is <math>\fbox{E}</math>. |
Latest revision as of 18:24, 17 October 2024
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 AA, 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
.