Find the legs of a rectangle if one of the legs is 7 cm larger than the other, and the hypotenuse of this triangle is 13 cm.

Let the length of the smaller leg be equal to X cm, AC = X cm, then the length of the larger leg is (X + 7) cm.

Let’s apply the Pythagorean theorem.

AB ^ 2 = AC ^ 2 + BC ^ 2.

13 ^ 2 = X ^ 2 + (X + 7) 2.

169 = X ^ 2 + X ^ 2 + 14 * X + 49.

2 * X ^ 2 + 14 * X – 120 = 0.

X ^ 2 + 7 * X – 60 = 0.

Let’s solve the quadratic equation.

D = b ^ 2 – 4 * a * c = 7 ^ 2 – 4 * 1 * (-60) = 49 + 240 = 289.

X1 = (-7 – √289) / 2 * 1 = (-7 – 17) / 2 = -24 / 2 = -12. (Doesn’t fit because it’s less than 0)

X2 = (-7 + √289) / 2 * 1 = (-7 + 17) 2 = 10/2 = 5 cm.

AC = 5 cm.

BC = 5 + 7 = 12 cm.

Answer: The legs of the triangle are 5 cm and 7 cm.