29
Proof of the Pythagorean theorem
Textbook: pp. 96–97
GoalUnderstand the area proof of the Pythagorean theorem and be able to explain it.
New words
proof · isbotsquares of equal area · tengdosh kvadratlarsum of the legs · katetlar yig‘indisifour congruent triangles · to‘rtta teng uchburchak
Explanation
Take a right triangle with legs a, b and hypotenuse c, and draw two identical squares of side a + b. Cut the first into four copies of the triangle and a middle quadrilateral of side c; its four sides equal c and its angle is 180° − (sum of the acute angles) = 90°, so it is a square. Cut the second into four more such triangles and two squares with sides a and b. Both big squares have the same area, so 4S + c² = 4S + a² + b², where S is the area of one triangle. Subtracting 4S from both sides gives c² = a² + b².
Worked examples
a = 3, b = 4: the big square has area 7² = 49 and the four triangles have 4 · 6 = 24, so c² = 49 − 24 = 25.
a = 5, b = 12: the big square is 17² = 289, the triangles are 4 · 30 = 120, c² = 169, c = 13.
Class activity
“Rebuilding the square”: on squared paper draw a square of side 7 cells and try to fill it in two ways with four triangles with legs 3 and 4.
Practice
1
The legs are 6 and 8. What is the area of the big square (a + b)²?
196
2
The legs are 6 and 8. What is the total area of the four triangles?
96
3
The legs are 6 and 8. By the proof, what is c²?
100
4
Why is the middle quadrilateral in the first drawing a square?
Its four sides equal c and its angle is 180° − 90° = 90°; a rhombus with a right angle is a square.