Given a rectangular trapezoid ABCD with bases AD = 14 cm, BC = 8 cm, in which cosC = 0.6. Find the area of the trapezoid.

Given: ABCD – rectangular trapezoid; AD, BC – bases; AD = 14 cm, BC = 8 cm, cos C = 0.6.
Find: S abcd -?
Decision.
Let’s draw a drawing for the task in accordance with the drawing in the photo.
1) Draw the height CH.
ABCH – rectangle => BC = AH = 8 cm.
HD = AD – AH = 14 cm – 8 cm = 6 cm.
2) Let ∠HCD = a.
Then ∠D = 180 ° – ∠CHD – ∠HCD = 180 ° – 90 ° – a = 90 ° – a.
According to the reduction formula: cos (90 – a) = sin a = 0, 6 = cos D = HD / CD.
HD / CD = 0.6 = 6/10.
HD = 6 cm, CD = 10 cm.
By the Pythagorean theorem:
CH ^ 2 = CD ^ 2 – HD ^ 2 = 10 ^ 2 – 6 ^ 2 = 100 – 36 = 64.
CH = 8 cm.
S trapezoid = 1/2 × h × (a + b).
S abcd = 1/2 × CH × (BC + AD) = 1/2 × 8 cm × (8 cm + 14 cm) = 4 cm × 22 cm = 88 cm.
Answer: 88 cm.