Question

A triangle and a parallelogram have the same base and the same area. If the sides of the triangle are 9 cm, 28 cm, and 35 cm and the parallelogram stands on the base is 9 cm, then the height (in cm) of the parallelogram is

• A.
  $3\sqrt{6}$
• B.
  $6\sqrt{3}$
• C.
  $4\sqrt{3}$
• D.
  $4\sqrt{6}$

Answer 1

The correct option is D

$4\sqrt{6}$
Let $\Delta ABC$and parallelogram PQCB have the same base BC and the same area.

Semi-perimeter, (s) of$\Delta ABC=\frac{35+28+9}{2}=36cm\therefore \text{Area of}\Delta ABC=\sqrt{s(s-a)(s-b)(s-c)}\text{(Using Heron's formula)}=\sqrt{36(36-9)(36-28)(36-35)}=\sqrt{36\times 27\times 8\times 1}=\sqrt{4\times 9\times 3\times 9\times 2\times 4}=36\sqrt{6}c{m}^{2}Given:Ar\left(PQCB\right)=Ar\left(\Delta ABC\right)\Rightarrow BC\times \text{Height of the parallelogram (h)}=36\sqrt{6}c{m}^2\Rightarrow 9\times h=36\sqrt{6}\Rightarrow h=4\sqrt{6}cm$

Hence, the correct answer is option (d).