LimpideTOOLS · VERIFIED EXERCISES

Pythagoras' theorem.

A formula everyone knows. One figure shows why it is true, and which side is c.

THEOREM · RIGHT TRIANGLETRAP · b² ≠ a² + c²

In a right triangle with hypotenuse c:

a2+b2=c2a^{2} + b^{2} = c^{2}
FIGURE · FOUR TRIANGLES, TWO ARRANGEMENTSSame square, same triangles: the same area is left.
Two squares with side a + b, each holding four copies of the right triangle with legs a and b and hypotenuse c. On the left, the triangles fill the corners and leave in the middle the square c², built on the hypotenuse. On the right, they make two rectangles ab and leave the squares a² and b², built on the legs.c²cabbaa²b²abab

WHY IT IS TRUE

  1. On the left, the four triangles fill the corners of the square with side a + b. The two acute angles of a right triangle add up to 90°, so the space left in the middle is a square, built on the hypotenuse:

    (a+b)2=4⋅ab2+c2{(a + b)}^{2} = 4 \cdot \frac{ab}{2} + c^{2}

  2. On the right, the same triangles, put together in pairs as rectangles, leave the squares built on the legs:

    (a+b)2=4⋅ab2+a2+b2{(a + b)}^{2} = 4 \cdot \frac{ab}{2} + a^{2} + b^{2}

  3. The same square, less the same four triangles: what is left has the same area.

    c2=a2+b2c^{2} = a^{2} + b^{2}

    The trap: c is not any side, it is the hypotenuse, opposite the right angle. To find a leg, subtract: b² = c² − a².

WITH NUMBERS

b2=132−52=169−25=144=122b^{2} = 13^{2} - 5^{2} = 169 - 25 = 144 = 12^{2}

c = 13 is the hypotenuse and a = 5 a leg: the unknown b is the other leg, so subtract, and b = 12.