Question

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

Answer 1

Let's find the area of triangle first :

Semi Perimeter of the triangle =

(26+28+30)/2 = 84/2 = 42 cm

Then ,

Area =

$\sqrt{s(s - a)(s - b)(s - c)} \\ \\ \sqrt{42(42 - 26)(42 - 28)(42 - 30)} \\ \\ \sqrt{42 \times 16 \times 14 \times 12} \\ \\ = 336 \: {cm}^{2}$

The area of the triangle = 336 cm²

Then it is given that the parallelogram and the triangle have same area . Also , the parallelogram is standing on the side 28 cm . Using this information let us find out the height:

Base × Height = Area of Parallelogram

28 × Height = 336

Height = 336/28 = 12 cm

The required height is 12 cm .

Answer 2

$\bf\huge\textbf{\underline{\underline{According\:to\:the\:Question}}}$

a = 26 cm, b = 28 cm and c = 30 cm.  

Perimeter of the triangle.  

$\bf\huge{\implies s = \dfrac{a+b+c}{2}}$          

$\bf\huge{\implies s = \dfrac{26+28+30}{2}}$          

$\bf\huge{\implies s = \dfrac{84}{2}}$          

= 42 cm  

Heron’s formula

$\bf\huge{\implies Area\:of\:Triangle = \sqrt{s(s-a)(s-b)(s-c)}}$          

$\bf\huge{\implies Area\:of\:Triangle = \sqrt{42(42-26)(46-28)(46-30)}}$          

$\bf\huge{\implies Area\:of\:Triangle = \sqrt{42\times 16\times 14\times 12}}$          

$\bf\huge{\implies Area\:of\:Triangle = \sqrt{7\times 6\times 16\times 2\times 7\times 6\times 2}$          

$\bf\huge{\implies Area\:of\:Triangle = \sqrt{7\times 7\times 6\times 6\times 16\times 2\times 2}$          

⇒ 7 × 6 × 4 × 2 = 336 cm²  

Area of parallelogram = base × height

⇒ 336 = 28 × h

$\bf\huge{\implies\dfrac{336}{28} = h}$          

⇒ 12 cm = h

$\bf\huge\bf\huge{\boxed{\bigstar{{Height \:= 12 \:cm}}}}$