Pythagorean Theorem


1)    What information is needed for you to calculate the area of the square?

2)    How can the two triangle measurements help you figure out the area of the square?

3)    Figure ADE is a right triangle. Where is the right angle? Where is the hypotenuse?

4)    How can you use the Pythagorean Theorem to figure out the length of line segment AD?

5)    What is the length of line segment AD? What is the length of line segment AB?

6)    What is the area of the square?

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Figure ABCD is a square. Figure ADE is a right triangle. What is the area of the square?

1) What information is needed to calculate the perimeter of the triangle? What information do you already have?

2) How can you figure out the length of line segment AD? You are given the area of square ABCD; how can that help you discover the length of AD?

3) How is the area of a square or rectangle usually calculated? How can you calculate this area, if you don’t know the length of any of the sides?

4) But the area has already been calculated. Can you think backwards to figure out what the length of the sides must be?  Are the sides all the same length? How do you know for sure? What number times itself equals 4? What is the square root of 4 square meters? What is the length of AD?

5) How can you figure out the length of line segment AE? You know the length of AD and DE, which are two sides of the triangle. What kind of triangle is it? Which side of the right triangle is the hypotenuse? If this is a right triangle, can you use the Pythagorean Theorem to calculate the length of the hypotenuse? What is the Pythagorean Theorem: Is it a + b = c? Is it a² + b² = c²? Does the Pythagorean Theorem tell you the length of the hypotenuse, or the length of the hypotenuse squared?

6) Extra hint for students who already know about 3-4-5 right triangles: double the measurements of AD and DE; what do you notice?

Figure ABCD is a square with an area of 4 square meters. Figure ADE is a right triangle. One side measures 1.5 meters as indicated. What is the perimeter of the triangle?  What prior concepts or procedures do students need in order to solve this problem?

1. Look at Figure #1.What is the length of the top? What is the length of the bottom? What is the width of the right side? What is the width of the left side?
2. Is Figure #1 a square?
3. Is Figure #2 exactly the same size and shape as Figure #1?
4. How many triangles are in Figure #1? Do they all have the same measurements? Are they all the same size? Are they all right triangles?
5. How many right triangles are in Figure #2? Are they all the same size as the triangles in Figure #1?
6. If the upper left triangle from each figure were removed, would the areas of the two figures still be equal?
7. If two triangles were removed from each figure, would the remaining areas still be equal?
8. If all of the triangles were removed from each figure, would the remaining areas still be equal?
9. How long is the shortest leg of each triangle? How long is the hypotenuse? How long is the other leg?
10. What are the dimensions of the smaller square in Figure #1? What is its area? Compare that square to one of the triangles; one side of the square is the same length as which leg of the triangle? Compare the length of that leg to the area of the small square.
11. What are the dimensions of the larger square in Figure #1? What is its area? Compare that square to one of the triangles; one side of the square is the same length as which leg of the triangle? Compare the length of that leg to the area of the larger square.
12. What are the dimensions of the large square in Figure #2? What is its area? Compare that square to one of the triangles; one side of the square is the same length as which leg of the triangle? Compare the length of that leg to the area of the large square.
13. If all of the triangles were removed from Figure #1, what would remain?
14. If all of the triangles were removed from Figure #2, what would remain? (8.) If all of the triangles were removed from each figure, would the remaining areas still be equal?
15. If you combined (added) the areas of the small square and the larger square in Figure #1, would that combined area (sum) be equal to the area of the large square in Figure #2?

And a tribute to someone with a brain, who could have used a few extra problems of the week on the Pythagorean Theorem. Ask your students if the diploma helped the scarecrow. Upon receiving his diploma he recited: “The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side. Oh, joy, oh, rapture. I’ve got a brain!”

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To wrap up this series of Pythagorean Problems of the Week, here is one that often presents difficulties for students at first. Where are the numbers?! What is the length of a side that is only assigned a letter!!?

There has been a purposeful flow to our weekly Pythagorean problems. We began with whole number measurements and solutions, the second one using the information to solve another problem, and moved to whole number measurements with an irrational number solution. We now end with a problem that preserves complete generality.

Can you use the “measurements” in these two pictures to prove that a2 + b2 = c2 is true for all right triangles? The length of “a” is the same in both figures, as is the length of “b” and “c.” For this problem, all corners that look square are square. The solution will be posted next week with a special tribute to the scarecrow from the Wizard of Oz.

1. How many degrees is angle ADC?
2. What kind of triangle is triangle ADC?
3. How many degrees is angle DAC?  …angle DCA?
4. How many degrees is angle ABC?
5. How many degrees is angle CAB?  …angle ACB?
6. How many degrees is angle  BAD?  …angle BCD?
7. Is figure ABCD a square?
8. What is the area of figure ABCD?
9. What is the area of triangle ABC?
10. Are triangles ABC, CBF, FBE, and EBA all the same size?
11. What is the area of figure EFCA?
12. Is figure EFCA a square?
13. What is the area of figure EFCA?
14. What is the measurement of each side of figure EFCA?
15. What is the length of  line segment AC?

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