Two circles, C1 and C2, with respective radii of 13 cm and 17 cm and their centres 20 cm apart, intersect each other at two points P and Q. Find the area (in sq.cm) of the biggest triangle that can be drawn in the region where the two circles overlap, such that two of the vertices of the triangle are P and Q.

To solve this problem, we need to understand that the largest triangle that can be formed in the overlapping region of two circles is when the third vertex of the triangle is at the intersection of the line joining the centres of the two circles and the line perpendicular to it passing through the point of intersection of the two circles.

Step 1: Find the distance of the third vertex from the line PQ.

The distance of the third vertex from the line PQ can be found using the formula for the distance between two points. The distance between the centres of the two circles is 20 cm. The radii of the two circles are 13 cm and 17 cm. Therefore, the distance of the third vertex from the line PQ is (20^2 - (17-13)^2)^0.5 / 2 = 12 cm.

Step 2: Find the length of the line segment PQ.

The length of the line segment PQ can be found using the formula for the distance between two points. The radii of the two circles are 13 cm and 17 cm. Therefore, the length of the line segment PQ is 2 * (17*13)^0.5 = 24 cm.

Step 3: Find the area of the triangle.

The area of the triangle can be found using the formula for the area of a triangle, which is 1/2 * base * height. The base of the triangle is the length of the line segment PQ, which is 24 cm. The height of the triangle is the distance of the third vertex from the line PQ, which is 12 cm. Therefore, the area of the triangle is 1/2 * 24 cm * 12 cm = 144 sq.cm.