Determining if a Long Object Will Fit around a Hallway Corner
Here’s an application of trigonometry that you may be able to relate to: Have you ever tried to get a large piece of furniture around a corner in a house? You twist and turn and put it up on end, but to no avail.
In this example, pretend that you’re trying to get a 15-foot ladder around a corner where two 4-foot-wide hallways meet at a 90-degree angle. The following figure shows a picture of the situation.
The tightest part comes when the ladder is halfway through the hallway, or when the angles where it touches the outer walls are the same. When the ladder is at the tightest point, it’ll form a right triangle with equal sides — half the ladder to each side of the corner.
Because the sides of the right triangle are equal at this point, you’ve got an isosceles right triangle, which has two 45-degree angles (see the preceding figure, which shows the longest a ladder can be to fit around the corner). How long are the sides of the right triangle, then? When you know the dimensions of this isosceles right triangle, you can look at the hypotenuse — the ladder — and determine if it’s short enough or too long to fit around the tightest part of this corner. And, of course, you don’t want to scrape or punch holes in the wall!
Determine the trig function that you can use with the measures available.All the triangles have the same angle measures.
The hypotenuse is the length of the ladder — 15 feet. The opposite and adjacent sides are the same in an isosceles right triangle, and in this case, those two lengths are each 8 feet. You know this measure because all the triangles are isosceles right triangles — which means they have 45-degree angles and equal leg measures (see the figure).
Determine which trig function to use.
Both sine and cosine include the length of the hypotenuse, which is what you’re solving for, so you can use either function.
Write the equation with the trig function, then input the measures that you know.
Solve for the value of the hypotenuse.
You find that at the tightest point around the corner, the hypotenuse is only slightly more than 11 feet. That 15-foot ladder will never fit around the corner. You need a shorter ladder!