The sides of the trapezoid are 13 cm and 15 cm, and the base lengths are 1: 3.

The sides of the trapezoid are 13 cm and 15 cm, and the base lengths are 1: 3. Find the area of a trapezoid if it is known that a circle can be inscribed into it.

By condition, a trapezoid is given: a – smaller base, b – larger base, c = 13 and d = 15 – lateral sides.
1. It is known from the condition that a circle can be inscribed into a given trapezoid. It is known from the properties of a trapezoid that if a circle can be inscribed into a trapezoid, then the sum of the bases of the trapezoid is equal to the sum of its lateral sides.
Let x be the coefficient of proportionality, then we denote the smaller base a as x, and the larger base b as 3x (because a: b = 1: 3). The expression turns out:
x + 3x = 13 + 15;
4x = 28;
x = 28/4;
x = 7 cm.
A smaller base a = x = 7 cm, then a larger base b = 3x = 3 * 7 = 21 (cm).
2. The height of the trapezoid is found by the formula:
h = √ (c ^ 2 – ¼ ((c ^ 2 – d ^ 2) / (b – a) + b – a) ^ 2)
Let’s substitute the values ​​we know into this formula:
h = √ (13 ^ 2 – ¼ ((13 ^ 2 – 15 ^ 2) / (21 – 7) + 21 – 7) ^ 2) = √ (169 – ¼ ((169 – 225) / 14 + 14) ^ 2) = √ (169 – ¼ ((- 56) / 14 + 14) ^ 2) = √ (169 – ¼ (- 4 + 14) ^ 2) = √ (169 – ¼ (10) ^ 2) = √ (169 – ¼ * 100) = √ (169 – 25) = √144 = 12 (cm).
3. Find the length of the midline of the trapezoid:
m = (a + b) / 2;
m = (7 + 21) / 2 = 28/2 = 14 (cm).
The area of ​​the trapezoid is found by the formula:
S = m * h;
S = 14 * 12 = 168 (cm square).
Answer: S = 168 cm square.