Find the area of a trapezoid in which: the bases are 8 cm and 23 cm, and the sides are 13 cm and 14 cm.

1. Let us be given a trapezoid with sides a, b, c, d, where the bases are a = 8 cm, b = 23 cm and the sides are c = 13 cm, d = 14 cm.

2. Let’s find the height of the trapezoid by the formula:

h = √ [c² – (((a – b) ² + c² – d²) / (2 (a – b)) ²] = √ [13² – (((23 – 8) ² + 13² – 14²) / ( 2 (8 – 23)) ²] = √ [169- ((15² + 169 – 196) / (2 * (- 15)) ²] = √ [169- ((225 + 169 – 196) / (- 30 )) ²] = √ [169- (198 / (- 30)) ²] = √ [169- 43.56] = √125.44 = 11/2 cm;

3. Find the area of the trapezoid knowing the length of the two bases and the height:

S = h * (a + b) / 2;

S = 11.2 * (8 + 23) / 2 = 11.2 * 31/2 = 173.6 cm²;

Answer: the area of the trapezoid is 173.6 cm².