Find the smaller base of the isosceles trapezoid if the larger base of the trapezoid is 25.

Find the smaller base of the isosceles trapezoid if the larger base of the trapezoid is 25. The height of the trapezoid is 8. The tangent of the acute angle Φ is 0.8

Let’s call our trapezium ABCD, where AB and CD are the sides of the isosceles trapezoid, BC and AD are the smaller and larger bases, respectively. Let’s draw the heights BL and CM, we get two identical right-angled triangles, consider the triangle BLA:
tgBAL = BL / AL, it follows that
AL = BL / tgBAL.
AL = 8 / 0.8 = 10 cm.
Since the lengths AL and MD are equal, then we find the length of the smaller base:
BC = AD – AL – MD;
BC = 25 – 2 * 10 = 5 cm.
Answer: the length of the smaller base of an isosceles trapezoid is 5 cm.