Problem 377

If the sides of a triangle have lengths 30,40 and 50 , what is the length of the shortest altitude?
(A) 20
(B) 24
(C) 25
(D) 30
(E) 40

## Solution

Since $30^{2}+40^{2}=50^{2}$ this is a right-angled triangle with a hypotenuse of 50 units.

The area of the triangle is $\frac{30 \times 40}{2}$ or 600 sq. units.

If we draw a perpendicular from the right angle and call this height, $h$, an expression for the area is $\frac{1}{2}(h)(50)=25 h$.

Equating the two, we have $25 h=600$ or $h=24$ which is the length of the shortest altitude.