| The legs of a right triangle are in the ratio 3:7, and the height dropped to the hypotenuse is 42. Find the segments into which the height divides the hypotenuse. 

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The product of the specified height and the hypotenuse equals the product of the legs.

## Solution

Let in triangle $A B C \quad \angle C=90^{\circ}$, the specified height $C D=42, B C: A C=3: 7$. Then $B D: A D=B C^{2}: A C^{2}=$ $3^{2}: 7^{2}$. Since $B D \cdot A D=h^{2}=2^{2} \cdot 3^{2} \cdot 7^{2}$, then $B D=2 \cdot 3^{2}, A D=2 \cdot 7^{2}=98$.

## Answer

18 and 98.