A length of road stretches north
for 10 kilometers from the edge of a town to where it intersects an eastward
road. A car is broken down on the
eastward road, and the displacement of the edge of the town from the car has a
magnitude of 24 kilometers. How far east of the intersection is
the car, to the nearest kilometer?
We’ll label this distance 𝐷, and
we’ll start off by drawing a diagram of this situation. We’re told that we have a town with
a road from the eastern edge of the town that moves up north. This northward-pointing road
intersects another road, which allows travel to the east. Along this east–west road, at some
location, there is a car broken down.
We’re told that the distance
between the broken down car and the town is 24 kilometers. And we’re also told that the
stretch of road that goes to the north before it intersects the eastern road is 10
kilometers. Given this information, we want to
solve for 𝐷, which is the distance from the broken down car to the
northward-pointing road along the eastward road.
Since the two roads are
perpendicular to one another, what we have is a right triangle with a hypotenuse of
24 kilometers and one leg of 10 kilometers. We can recall the Pythagorean
theorem, which says that, for a right triangle with sides 𝐴 and 𝐵 and hypotenuse
𝐶, the length 𝐶 squared is equal to the length 𝐴 squared plus the length 𝐵
Applying this theorem to our
scenario, we can say that 24 kilometers squared is equal to 10 kilometers squared
plus 𝐷 squared, or that 𝐷 is equal to the square root of 24 kilometers squared
minus 10 kilometers squared. Entering this expression on our
calculator, we find, to the nearest kilometer, that 𝐷 is 22 kilometers. That’s the distance along the
eastward road that the broken down car is from the northward-pointing road.