A diagonals of a parallelogram measure 16cm and 14 cm respectively. The shorter side measures 10cm. find the longer side.

Draw ||gm ABCD, with AB and CD as the short sides and angle ABC obtuse. Draw diagonal BD = 14 cm and diagonal AC = 16 cm. Let E be the point of intersection of the two diagonals.

Diagonals of a ||gm bisect each other, so BE = ED = 7 and AE = EC = 8
It is also given that AB = CD = 10.
In triangle ABE, use the cos law to find angle BEA.
In triangle BCE, use the fact that angle BEC is supplementary to angle BEA to find its value. You now have enough information to find side BC, using the cos law.

To find the longer side of the parallelogram, we can use the concept of the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

In a parallelogram, the diagonals bisect each other, forming two congruent right triangles. Let's consider one of those triangles.

Let the shorter side of the parallelogram be the base of the right triangle, measuring 10 cm. Let the lengths of the two diagonals be the other two sides of the right triangle.

According to the given information, one of the diagonals measures 16 cm, and the other measures 14 cm.

Using the Pythagorean theorem, we can calculate the length of the longer side as follows:

Longer side^2 = (Half of a diagonal)^2 + (Half of a diagonal)^2
Longer side^2 = (16/2)^2 + (14/2)^2
Longer side^2 = 8^2 + 7^2
Longer side^2 = 64 + 49
Longer side^2 = 113

To find the length of the longer side, we take the square root of both sides:

Longer side = √113 cm

So, the longer side of the parallelogram measures approximately 10.63 cm.