Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.5 AIME Prove it United States

Problem:

$AD$ and $BC$ are both perpendicular to $AB$ , and $CD$ is perpendicular to $AC$ . If $AB = 4$ and $BC = 3$ , find $CD$ .

Solution

Solution:

By Pythagoras in $\triangle ABC$ , $AC = 5$ . But $\angle CAD = 90^{\circ} - \angle BAC = \angle ACB$ , so right triangles $CAD$ and $BCA$ are similar, and $\frac{CD}{AC} = \frac{BA}{CB} = \frac{4}{3} \Rightarrow CD = \frac{20}{3}$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.