In a rectangular trapezoid, the bases are 13 and 22, and the large side is 15. Find the smaller side.

Given: ABCD – rectangular trapezoid;
ВС and АD – bases;
lateral side CD = 15;
BC = 13;
AD = 22;
Find the length AB -?
Decision:
1) Consider a rectangular trapezoid ABCD. Let’s draw the height of the CК. Then AD = Ak + CD; KD = AD – AK;
CD = 22 – 13 = 9;
2) The quadrilateral ABCK is a rectangle. Therefore BC = AK = 13;
AB = SK;
3) Consider a right-angled triangle CKD. By the Pythagorean theorem, CK ^ 2 = CD ^ 2 – KD ^ 2;
CK ^ 2 = 15 ^ 2 – 9 ^ 2 = 225 – 81 = 144;
CK = 12;
Then CK = AB = 12.
Answer: 12.