Why the theorem holds

The idea

Marking a rectangle on the larger square with a side of the smaller and drawing four congruent right triangles of sides a, b gives a square on the hypotenuse whose area equals the two given squares together.

Why does it work?

Last lesson we measured the big square on the hypotenuse. Now we will see why its area is the area of the two squares together.

Stand a square of side a and a square of side b side by side. Cut off two right triangles with perpendicular sides a and b: one purple, one orange. Then move them to new places.

What do you think the pieces will make?

• a long rectangle
• a tilted square
• a bigger triangle

Turn and slide

Tap a triangle, or press its button. It turns and slides into its dashed place. Move both triangles. Then try a second pair of squares.

Same pieces, new shape

Move two right triangles (perpendicular sides a and b) to new places. The two squares become one square on the hypotenuse. Nothing is added and nothing is lost, so the areas are equal.

Why is it a square? Each side is the hypotenuse of a triangle with perpendicular sides a and b. The four triangles are all the same, so all 4 sides are equal.

And each corner is a right angle: angles along a straight line add to 180°, and the two pointed angles of the triangle add to 90°, so the corner is left with 90°.

Notes

Why it works. Join squares of sides a and b. Cut off two right triangles (perpendicular sides a and b), turn them and slide them. The pieces make one square on the hypotenuse: nothing added, nothing lost, so the areas are equal.

Check yourself

Two triangles move to new places. What do you know about the area of the new square?

The squares have sides 3 and 4, so together their area is 25. Each triangle has area 6 (half of 3 × 4). The two triangles are cut off and the middle piece V stays. What is the area of V?

Answer: 13

25 − 6 − 6 = 13. The new square is V and the two moved triangles: 13 + 6 + 6 = 25.

Why are all four sides of the new figure equal?

Squares of sides 2 and 5 are joined, and the two triangles are moved. What is the area of the new square?

Answer: 29

2 × 2 + 5 × 5 = 4 + 25 = 29. Its side is a little more than 5, because 5 × 5 = 25 and 6 × 6 = 36.

• It is bigger, because the triangles now cover new space.. No new space is covered. The triangles only move; they do not grow.
• It is the same as the two squares together, because the pieces only moved. — correct. Yes! The pieces only moved. Nothing was added and nothing was lost.
• It is smaller, because the triangles left their old places.. The old places are empty, but the triangles fill new places of the same size. The total area is unchanged.
• Because the two squares we started with were equal.. The squares had different sides, a and b. That is not the reason.
• Each side is the hypotenuse of a triangle with sides a and b, and the triangles are all the same. — correct. Yes! Four sides are four hypotenuses of the same triangle.
• Because the figure is tilted.. Tilting does not change any length. The sides are equal because the triangles are the same.