The height drawn from the top of an obtuse angle of an isosceles trapezoid divides the larger base of the trapezoid

The height drawn from the top of an obtuse angle of an isosceles trapezoid divides the larger base of the trapezoid into two segments, the smaller of which is 2 cm.Find the larger base of the trapezoid if its midline is 8 cm.

Given:
isosceles trapezoid ABCE,
BК – height,
AK = 2 centimeters,
NM is the middle line.
Find the length of the larger base, that is, AE -?
Solution:
1. Let’s draw the CO height. Right-angled triangle ABK = right-angled triangle COE in hypotenuse and acute angle, since AB = CE and angle A = angle E because trapezoid ABCE is isosceles. Then AK = OE = 2 centimeters;
2. The quadrilateral BCOK is a rectangle.
3. Let BC = x centimeters, then AE = 2 + 2 + x centimeters. We know that the middle line is equal to the half-sum of the bases. Let’s make the equation:
(x + x + 2 + 2): 2 = 8;
x + x + 2 + 2 = 16;
x + x + 4 = 16;
x + x = 16 – 4;
x + x = 12;
x * (1 + 1) = 12;
x * 2 = 12;
x = 12: 2;
x = 6 centimeters – the lengths of the sides BC and KE;
AE = 2 + 2 + 6 = 10 centimeters – the length of the larger base of AE.
Answer: 10 centimeters.