Two adjacent sides of a parallelogram are 74 cm and 40 cm. If one diagonal is…

When a diagonal of a parallelogram is drawn, it divides the parallelogram into two triangles that are congruent to each other, since they share the diagonal and have equal pairs of opposite sides (SSS congruence). This means the parallelogram's area is exactly twice the area of either triangle. When all three sides of a triangle are known, Heron's formula gives its area directly: for a triangle with sides a, b and c and semi-perimeter s = (a + b + c) / 2, the area equals the square root of s(s minus a)(s minus b)(s minus c).

1. The diagonal of 102 cm, together with the two adjacent sides of 74 cm and 40 cm, forms a triangle with sides 74 cm, 40 cm and 102 cm.

2. Semi-perimeter: s = (74 + 40 + 102) / 2 = 108 cm.

3. s minus a = 108 minus 74 = 34 cm, s minus b = 108 minus 40 = 68 cm, and s minus c = 108 minus 102 = 6 cm.

4. Area of this triangle = the square root of (108 times 34 times 68 times 6) = the square root of 1,498,176 = 1224 sq cm.

5. The parallelogram is made up of two such congruent triangles, so its total area = 2 times 1224 sq cm = 2448 sq cm.

As a check: 74 cm + 40 cm = 114 cm, which is greater than the given diagonal of 102 cm, confirming that a valid triangle can be formed with these three lengths. Also, the total area of 2448 sq cm is below the highest area this parallelogram could have (74 times 40 = 2960 sq cm, reached only if the two given sides were perpendicular), which is consistent with a 102 cm diagonal indicating an angle other than a right angle between the sides.

So the area of the parallelogram is 2448 sq cm.