The perimeter of a 4-gon is 34, one of its sides is 10, and the other is 12.

The perimeter of a 4-gon is 34, one of its sides is 10, and the other is 12. Find the largest of the remaining sides of this 4-gon if you know that you can inscribe a circle into it.

If a circle can be inscribed into a triangle, then the sums of the lengths of its opposite sides are equal. Therefore, the sum of the lengths of each of the pairs of opposite sides is equal to half the perimeter and is equal to 34/2 = 17. Obviously, the sides equal to 10 and 12 are adjacent, since their sum is 22.

The remaining sides are equal:

17 – 10 = 7;

17 – 12 = 5.

The greater of the remaining sides is 7.