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.