Difference between revisions of "1962 AHSME Problems/Problem 23"
(Created page with "==Problem== In triangle <math>ABC</math>, <math>CD</math> is the altitude to <math>AB</math> and <math>AE</math> is the altitude to <math>BC</math>. If the lengths of <math>AB</m...") |
Erringbubble (talk | contribs) m (SSS is incorrect. We are not given any ratios. the correct one is AA, using the right angle and <B.) |
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<math>\textbf{(A)}\ \text{not determined by the information given} \qquad</math> | <math>\textbf{(A)}\ \text{not determined by the information given} \qquad</math> | ||
+ | |||
<math>\textbf{(B)}\ \text{determined only if A is an acute angle} \qquad</math> | <math>\textbf{(B)}\ \text{determined only if A is an acute angle} \qquad</math> | ||
+ | |||
<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> | ||
==Solution== | ==Solution== | ||
− | + | ||
+ | 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 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 .