Solution to the Hallway Problem

Label the vertices of the first window (trapezoid) A, B, C, and D. We want a second trapezoid WXYZ which is similar to the first and is drawn in perspective.

The segment AB is the top edge of the first window. The line AB passes through the vanishing point. The segment WX is the top edge of the second window and the line WX must also pass through the vanishing point. This is because the window tops are parallel. Since we choose to make the top edges of both windows the same height, the points A,B and W,X will all be on the same line which goes through the vanishing point.

The same is true for the bases of the windows.

The difficulty lies in finding the width of the second window. If trapezoid ABCD and trapezoid WXYZ are similar figures, what can you say about line AC and line WY? Why?

Line AC and line WY are

If you have access to a copy of The Geometer's Sketchpad(R), try using it to create a hallway with windows.

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