SAT Lines, angles, and triangles Question

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Question 1 · Hard · MathDecember 2025 Batch 3 · Q22Lines, angles, and triangles · 0 solved

In triangle ABC, the measure of angle B is $9 0^{\circ}$ and BD is an altitude of the triangle. The length of AB is 13 and the length of AC is 19 greater than the length of AB. What is the value of $\frac{B C}{B D}$ ?

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Strategy

In a right triangle, look for a relationship involving the altitude to the hypotenuse. If you know two sides, use the right-triangle altitude formula or a similar-triangles relation to rewrite the ratio you want in terms of those sides. Then substitute carefully and simplify.

Hints (open one at a time)

Hint 1

Since one angle is 90°, think of the triangle as a right triangle with a special altitude drawn to the hypotenuse.

Hint 2

A right triangle with an altitude to the hypotenuse gives a proportion linking the two legs, the altitude, and the hypotenuse.

Hint 3

Rewrite the requested ratio using that proportion, then replace the hypotenuse and leg lengths with the values you’re given.

Step-by-step Explanation

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