[ Projections of the bases, sides, or vertices of a trapezoid ] $[\quad$ The midline of a trapezoid $\quad]$

It is known that a circle can be inscribed in an isosceles trapezoid $A B C D$ with bases $A D>B C$; $C H-$ is the height of the trapezoid, $A H=7$. Find the lateral side of the trapezoid.
Isosceles trapezoid $A B C D$ is circumscribed around a circle, so $2 A B=A B+A B=A B+C D=A D+B C$, which means $A B=\frac{1}{2}(A D+B C)$. On the other hand, the projection of the diagonal of an isosceles trapezoid onto the larger base is equal to half the sum of the bases, i.e., $A H=\frac{1}{2}(A D+B C)$, and the half-sum of the bases of any trapezoid is equal to its midline. Therefore, the midline of trapezoid $A B C D$ is equal to the lateral side, i.e., 7.

## Answer

7.00

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