Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 3.9 AMC 10/12 Find the answer United States

Problem:

Points $A$ , $B$ , $C$ , and $D$ lie on a line in that order such that $\frac{AB}{BC}=\frac{DA}{CD}$ . If $AC=3$ and $BD=4$ , find $AD$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Solution:

Let $BC = x$ , then the equation becomes $\frac{3-x}{x}=\frac{7-x}{4-x}$ . This simplifies to a quadratic equation with solutions $x=1$ and $x=6$ . Since $x<3$ , we have $x=1$ and $AD=7-x=6$ .

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