Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 3.0 AMC 10/12 Find the answer

In triangle $ABC$ , $CD$ is the altitude to $AB$ and $AE$ is the altitude to $BC$ . If the lengths of $AB$ , $CD$ , and $AE$ are known, the length of $DB$ is:

Pick one

Solution

We can actually determine the length of $DB$ no matter what type of angles $A$ and $B$ 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 $DB$ if $A$ is an obtuse angle.
Let's see what happens when $A$ is an obtuse angle. $\triangle AEB\sim \triangle CDB$ by SSS, so $\frac{AE}{CD}=\frac{AB}{DB}$ . Hence $DB=\frac{AE}{CD\times AB}$ . Since we've determined the length of $DB$ even though we have an obtuse angle, $DB$ is not $\bf{only}$ determined by what type of angle $A$ may be. Hence our answer is $\fbox{E}$ .

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.