The middle line of the trapezoid divides its area in a ratio of 5: 7. find the ratio of the bases of the trapezoid.

Find the area of ​​the smaller trapezoid formed by drawing the middle line
The smaller base of this trapezoid is a, the larger base of this trapezoid is m, the height of this trapezoid is h / 2, therefore, the area S1 of this trapezoid is:

S1 = (a + m) / 2 * h / 2.

Find the area of ​​the larger trapezoid formed by drawing the middle line
The smaller base of this trapezoid is m, the larger base of this trapezoid is b, the height of this trapezoid is h / 2, therefore, the area S2 of this trapezoid is:

S2 = (m + b) / 2 * h / 2.

Find the ratio of the areas of these two trapezoids
S1 / S2 = ((a + m) / 2 * h / 2) / ((m + b) / 2 * h / 2) = (a + m) / (m + b).

Substituting the value m = (a + b) / 2 into this ratio, we get:

(a + m) / (m + b) = (a + (a + b) / 2) / ((a + b) / 2 + b) = ((2a + a + b)) / 2 / (( a + b + 2b) / 2) = (3a + b) / (a ​​+ 3b).

Divide the numerator and denominator of the resulting expression by b:

(3a + b) / (a ​​+ 3b) = ((3a + b) / b) / ((a + 3b) / b) = (3 (a / b) + 1) / (a ​​/ b + 3) …

Hence,

S1 / S2 = (3 (a / b) + 1) / (a ​​/ b + 3).

Find the ratio of the bases of the trapezoid
According to the condition of the problem, the middle line of the trapezoid divides its area in a ratio of 5: 7, therefore, we can write the following ratio:

(3 (a / b) + 1) / (a ​​/ b + 3) = 5/7.

Solving this equation for a / b, we get:

7 * (3 (a / b) + 1) = 5 * (a / b + 3);

21 (a / b) + 7 = 5 (a / b) + 15;

21 (a / b) – 5 (a / b) = 15 – 7;

16 (a / b) = 8;

a / b = 8/16;

a / b = 1/2.

Consequently, the bases of this trapezoid are related as 1: 2.

Answer: the bases of this trapezoid are related as 1: 2.