mathschallenge.net

Problem

A slide is made by joining a 6 foot ladder to an 8 foot chute at right angles.

Find the vertical height of the slide.

Solution

We shall begin by using the Pythagorean Theorem to calculate the base length of the slide.

$\therefore b^2 = 6^2 + 8^2 = 36 + 64 = 100 \implies b = 10$

By considering 6 to be the "base" and 8 to be the "height" of the triangle we can see that the area of the triangle is given by $\dfrac{6 \times 8}{2} 24$.

However, by considering the triangle in the current orientation, $A \Delta = \dfrac{bh}{2} = \dfrac{10h}{2} = 24 \implies 10h = 48 \implies h = \dfrac{24}{5}'$

(Difficult) Extension:

Given that the pair of lengths (6,8) produces a slide with a vertical height of 4.8, find the smallest pair of integer lengths that produce a slide with a vertical height that is also integer.