Compare your answers.

 

A rectangular pyramid has the measures shown, where all of the slant heights are the same length. When looked at from above, it appears that a + b = 180° and c + d = 90°, but neither is true.

a. Show that and that .

Labelling the sides may make answering this question easier.

Use the cosine law to determine angle c.

Angle c is part of an isosceles triangle.

Use the cosine law to determine angle d.

It is also the case that angle d is part of an isosceles triangle.

So,

So,

 

b. Explain the discrepancy.

When looking at the pyramid from above, you are looking at each triangle from an angle instead of straight on, which distorts the appearance of the angle measures. If you laid each triangle flat (imagine "unfolding" the pyramid from its base), you would have an image like the one below left. Here you can see the distortion more clearly, especially in comparison with the top view of the pyramid. And, it is clear that and .

 

For further information about solving problems in three dimensions see p. 159 of Principles of Mathematics 11.

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