A triangle and a parallelogram have the same base and the same area. If the sides of the triangle are 26 cm, 28 cm and 30 cm, and the parallelogram stands on the base 28 cm, find the height of the parallelogram.

Explanation

Given: Area of the parallelogram = Area of the triangle
By using the area of the parallelogram formula, we can calculate the height of the parallelogram
By using Heron’s formula, we can calculate the area of a triangle.
Heron's formula for the area of a triangle is: Area `= sqrt(s(s - a)(s - b)(s - c))`
Where a, b, and c are the sides of the triangle, and `s =` Semi-perimeter `=` Half the Perimeter of the triangle
Let `ABCD` is a parallelogram and `ABE` is a triangle having a common base with parallelogram `ABCD`.
For `triangle ABE, a = 30 cm, b = 26 cm, c = 28 cm`
Semi Perimeter: `(s) = Perimeter/2`
`s = (a + b + c)/2`
`= (30 + 26 + 28)/2`
`= 84/2`
`= 42 cm`
By using Heron’s formula,
Area of a` triangle ABE = sqrt(s(s - a)(s - b)(s - c))`
`= sqrt(42(42 - 30)(42 - 28)(42 - 26))`
`= sqrt(42 xx 12 xx 14 xx 16)`
`= 336 cm^2`
Area of parallelogram `ABCD =` Area of `triangle ABE` (given)
Base `xx` Height `= 336 cm^2`
`28 cm xx` Height `= 336 cm^2`
On rearranging, we get
Height `= 336/28 cm = 12 cm`
Thus, height of the parallelogram is `12 cm`.