Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 4.6 AIME Find the answer

Points $A, B, C$ , and $D$ lie on a line in that order such that $\frac{A B}{B C}=\frac{D A}{C D}$ . If $A C=3$ and $B D=4$ , find $A D$ .

Your answer

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

Solution

Let $B C=x$ , 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 $A D=7-x=6$ .

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