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.

Let the sides of the triangle be:

a = 26 cm, b = 28 cm and c = 30 cm.

The semi-perimeter s of the triangle is:

$∵ s = \dfrac{a + b + c}{2}\\[1em] = \dfrac{26 + 28 + 30}{2}\\[1em] = \dfrac{84}{2}\\[1em] = 42$

∵ Area of triangle = $\sqrt{s(s - a)(s - b)(s - c)}$

= $\sqrt{42(42 - 26)(42 - 28)(42 - 30)}$ cm²

= $\sqrt{42 \times 16 \times 14 \times 12}$ cm²

= $\sqrt{112,896}$ cm²

= 336 cm²

Area of triangle = Area of parallelogram

(∵ Area of parallelogram = base x height)

Let h be the height of parallelogram.

⇒ 336 = 28 x h

⇒ h = $\dfrac{336}{28}$

⇒ h = 12 cm

Hence, the height of the parallelogram is 12 cm.