Pythagorean Theorem

• Why should I bother learning this?
Ask students to imagine building a deck or constructing a quilt. How could they plan the layout and the cutting scheme so that the corners come out square? It turns out that ancient Egyptians used ropes knotted in 12 equal intervals. If three intervals and four intervals were laid along two edges of a supposedly square field and if the last five intervals reached diagonally across from one to the other, they were sure that corner was square.

• Why can't I just measure the missing side?
Measurement, because it uses a tool and the human eye, is never precise. Rulers vary and besides, it may be hard to read between the eighth-inch or millimeter marks. However, for the same reasons, computing may give you an answer that is too precise. When the answer to a problem is a measurement, the number of decimal places in the answer should really not be greater than the number of decimal places in the original measurements — the original measurements can't suddenly get more precise because you computed with them. However, MathSteps shows answers to Pythagorean Theorem problems rounded to the nearest tenth.

• Can I use the Pythagorean Theorem for triangles that do not have a right angle?
Not usually. The ratios among the sides of right triangles are special and the right angle is what makes these ratios work. No right angle means no special relationship. The only exception is when you are using the altitude of a non-right triangle in a problem.

This is an isosceles triangle. The altitude of any triangle is perpendicular to the base. This means that the altitude creates two right triangles. The altitude of an isosceles triangle is not only perpendicular, it also bisects the base. For this triangle, you have enough information to find the lengths of the congruent sides by using the Pythagorean Theorem (42 + 52 = 16 + 25, so the missing sides of this isosceles triangle are 41 units long).