Question

a triangle and parallelogram have same base and the same area if the sides of triangle are 26cm,28cm and 23cm and the parallelogram stands on the base 28cm, find the height of parallelogram ​

Answer 1

Given :

• Sides of the triangle :

⠀⠀⠀⠀⠀⠀⠀⠀⠀→ a = 26 cm

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀→ b = 28 cm

⠀⠀⠀⠀⠀⠀⠀→ c = 23 cm

• Base of the parallelogram = 28 cm

To find :

Height of Parallelogram.

Solution :

According to the Question , the area of the Parallelogram is equal to the area of the triangle , so by finding the area of the triangle and using it in the formula for area of a Parallelogram , we get :

Area of the triangle :

We know the heron's formula i.e,

$\underline{\bf{A = \sqrt{s(s - a)(s - b)(s - c)}}}$

Where :

• a,b, and c are the sides of the triangle.

• s = Semi-perimeter

Semi-Perimeter = $\bf{\dfrac{a + b + c}{2}}$

Let's first find the semi-perimeter i.e,

$:\implies \bf{s = \dfrac{26 + 28 + 23}{2}}$

$:\implies \bf{s = \dfrac{77}{2}}$

$:\implies \bf{s = 38.5}$

$\boxed{\therefore \bf{Semi-Perimeter\:(s) = 38.5\:cm}}$

Hence, the semi-perimeter is 38.5 cm.

Now Using the herons formula and substituting the values in it, we get :

$:\implies \bf{A = \sqrt{38.5(38.5 - 26)(38.5 - 28)(38.5 - 23)}}$

$:\implies \bf{A = \sqrt{38.5 \times 12.5 \times 10.5 \times 15.5}}$

$:\implies \bf{A = \sqrt{78323.44}}$

$:\implies \bf{A = 279.86}$

$\boxed{\therefore \bf{Area\:(A) = 279.86\:cm^{2}}}$

Hence, the area of the triangle is 279.86 cm².

Height of the Parallelogram :

Using the formula for area of a Parallelogram and substituting the values in it, we get :

$\boxed{:\implies \bf{A = Base \times height}}$

$:\implies \bf{279.86 = 28 \times height}$

$:\implies \bf{\dfrac{279.86}{28} = height}$

$:\implies \bf{9.99\:(approx.) = height}$

$:\implies \bf{10 = height}$

$\boxed{\therefore \bf{Height\:(h) = 10\:cm}}$

Hence, the height of the Parallelogram is 10 cm.