A circle is inscribed in a rectangular trapezoid. The tangency point divides the large lateral side into 8cm

A circle is inscribed in a rectangular trapezoid. The tangency point divides the large lateral side into 8cm and 18cm segments. Find the perimeter of the trapezoid.

Consider a trapezoid ABCD. By the condition of the problem:

point E is the point of contact between the trapezoid and the incircle and CE = 8 cm, ED = 18 cm.

Note that CF = CE = 8 and DE = DH = 18.

We have:

CD = 8 + 18 = 26, DH – CF = 18 – 8 = 10,

From the vertex C we drop the height SK and in the triangle CKD by the Pythagorean theorem we have

FH = CK = √CD ^ 2 – DK ^ 2 = √26 ^ 2 – 10 ^ 2 = 24 and the radius is 24/2 = 12.

Then the perimeter of the trapezoid is:

P = AB + BF + AH + HD + CD + CF = 24 + 12 + 12 + 18 + 26 +8 = 100.

Answer: 100.