Say:You know that the Pythagorean Theorem says that a^{2} + b^{2} = c^{2} for any right triangle, where a and b are the lengths of the legs and c is the length of the hypotenuse. This is a useful thing to know, because you can use it to find missing measures in a variety of situations. This is called indirect measurement.
Ask:If I told you that the length of this room is 25 feet and the width is 15 feet, could you find the distance from one corner to the opposite corner?
Students should spot this as a right triangle problem (hopefully, your room is rectangular; if it isn't, substitute another room that is). Have someone diagram the situation and label the information you know and what you need to find out. Then, have someone else use the Pythagorean Theorem to find the missing information. In this case 252 + 152 = 625 + 225, or 850. This means that, in the equation a^{2} + b^{2} = c^{2}, c^{2} = 850. Now, students may have some trouble figuring out what to do to find c.
Ask:What's another way to write c^{2}?
Talk about c^{2} as the area of a square whose sides are c units long. This means cc = c^{2}. So, in the case where c^{2} = 850, you need to find a number that you can multiply by itself to get 850.
Show students the radical notation c = 850. Then, depending on how you want students to solve problems like this, show them how to use the calculator or table of squares and square roots and/or how to simplify under the radical and leave their answers in radical form. In this case, c 29.154 or c = 534. The answer to the original problem would be The distance from one corner to the opposite corner of the room is 534 feet or about 29 feet.
Continue the lesson with other problems of this type:
You're building a deck on the back of your house. It's 12 feet long and 5 feet wide. How can you be sure you get the corners square? Solution: If each diagonal is 13 feet long, your corners are square.
You're making a quilt and one square block is made of triangles like this:
If you made each triangle by cutting a 6-inch square along the diagonal, what are the approximate dimensions of the quilt block? Ignore seam allowances.
One way to solve: If each small triangle has legs 6 inches long, then the diagonal of the block is 12 inches long. Since we know the block is a square, we can modify the Pythagorean Theorem to a^{2} + b^{2} = c^{2}, or 2a^{2} = 144. This means that a^{2} = 72 and the side of the block is about 8.5 inches long.
Another way to solve: If each small triangle was made by cutting a 6-inch square along the diagonal, then each side of the block is the hypotenuse of a right triangle whose legs are both 6 inches long. 6^{2} + 6^{2} = c^{2}. 72 = c^{2}, and the side of the block is about 8.5 inches long.
Wrap-Up and Assessment Hints
The Pythagorean Theorem is a very useful tool, but the computation can frighten students unless you give them lots of practice with:
recognizing perfect squares;
recognizing common Pythagorean triples;
and using a calculator or a table of square roots.