The Pythagorean Theorem Problem 3.1

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Presentation transcript:

The Pythagorean Theorem Problem 3.1 The longest side of a right triangle is the side opposite the right angle. We call this side the HYPOTENUSE of the triangle. The other two sides are called LEGS. LEG HYPOTENUSE

Consider a right triangle with legs that each have a length of 1 Consider a right triangle with legs that each have a length of 1. Suppose you draw squares on the hypotenuse and the legs of the triangle. How are the areas of these three squares related?

Draw the required squares on your dot paper and complete the chart Draw the required squares on your dot paper and complete the chart. Look for patterns.

Draw the required squares on your dot paper and complete the chart Draw the required squares on your dot paper and complete the chart. Look for patterns.

Draw the required squares on your dot paper and complete the chart Draw the required squares on your dot paper and complete the chart. Look for patterns.

Draw the required squares on your dot paper and complete the chart Draw the required squares on your dot paper and complete the chart. Look for patterns.

Draw the required squares on your dot paper and complete the chart Draw the required squares on your dot paper and complete the chart. Look for patterns.

Area of Hypotenuse square C. Look for patterns in the relationship among the areas of the three squares drawn for each triangle. Use the pattern you discover to make a conjecture (hypothesis) about the relationship among the areas. Area of Leg 1 square Area of Leg 2 Square Area of Hypotenuse square + =

Go to Geometer’s SketchPad D. Draw a right triangle with side lengths that are different from those given in the table. Use your triangle to test your conjecture from part C. Go to Geometer’s SketchPad

Now that I know the area of the Hypotenuse Square, I can I find the one thing that I don’t know about a right triangle? 40 sq units 36 sq units 6 2 4 sq units

3.1 Follow Up: Record the length of the hypotenuse using the square root method. Approximate your answer to the nearest hundredth.

Here is the importance of what you just learned: You were given the lengths of the two LEGS and were asked to find the HYPOTENUSE. You found that if you found the areas of the two squares formed from legs and added them up, it equaled the area of the square formed from the hypotenuse. Then you found the length of the hypotenuse by taking the square root of that larger area. Try these two problems. Find the length of the hypotenuse for each. Leg 4 . Leg 2 . Hypotenuse . Leg 5 . Leg 6 . Hypotenuse .

Pythagoras wrote his theorem as: a2 + b2 = c2 4.47 Leg 4 . Leg 2 . Hypotenuse . 4 x 4 = 16 2 x 2 = 4 16 + 4 = 20 20 = 4.47 7.81 Leg 5 . Leg 6 . Hypotenuse . 5 x 5 = 25 6 x 6 = 36 25 + 36 = 61 61 = 7.81 Pythagoras wrote his theorem as: a2 + b2 = c2 It only matters that “c” is in the spot of the hypotenuse in the formula. It does not matter which leg you make “a” and which leg you make “b”. b a c

You just proved this: a2 + b2 = c2

If you knew the hypotenuse and one of the legs could you find the other? Leg 9 . Leg ? . Hypotenuse 15 . a2 + b2 = c2 92 + b2 = 152 81+ b2 = 225 b2 = 144 b2 = 144 = 12