# The larger base of the trapezoid is 8. A straight line parallel to the bases divides the trapezoid

The larger base of the trapezoid is 8. A straight line parallel to the bases divides the trapezoid into two equal-sized figures. The segment of this straight line inside the trapezoid is 5√2 Find the smaller base of the trapezoid.

From the top of C we draw the height CH and the straight line CP parallel to AB.

Let the length of the base BC = X cm.

Then the segment EM = (KM – X) cm, the segment PD = AD – BC = (AD – X) cm.

Triangles PCD and ECM are similar in two angles, then EM / PD = CT / CH.

(KM – X) / (AD – X) = CT / CH.

CT = CH * (KM – X) / (AD – X).

The area of ​​the trapezium VM is equal to:

Skvsm = (X + KM) * ST / 2.

The area of ​​the trapezoid ABCD is equal to:

Savsd = (X + AD) * CH / 2.

Since the straight line KM divides the trapezoid into two equal sizes, then 2 * Skvcm = Savsd.

2 * (X + KM) * ST / 2 = (X + AD) * CH / 2.

(X + KM) * CH * (KM – X) / (AD – X) = (X + AD) * CH / 2.

2 * (KM ^ 2 – X ^ 2) = X ^ 2 – AD ^ 2.

2 * (50 – X ^ 2) = 64 – X ^ 2.

100 – 64 = X ^ 2.

X ^ 2 = 36.

X = BC = 6 cm.

Answer: The length of the smaller base is 6 cm.

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