Question

A triangle and a parallelogram have the samebase and same area .If the sides of the triangle are 9cm,12cm and 15cm and the parallelogram stands on the base 9cm ,Find the height of the parallelogram
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Answer 1

ATQ,

The given statements are as under :-

The triangle and the parallelogram have same base, i.e., 9 cm.

Three sides of the Δ are given as under:-

Base = 9 cm

Others sides = 12 cm and 15 cm respectively.

Also,

Area of Δ = Area of llgm

which means,

Area of Δ = Base x  Height

Area of Δ = 9 x h

∴ h = 1/9 x ( Area of Δ ) ................ (eq.1)

Now,

Area of Δ = √s(s-a)(s-b)(s-c)

where a,b,c are the sides of the triangle and s is the semi-perimeter of the triangle.

s = (a+b+c)/2

  = (9+12+15)/2

  = 36/2

  = 18 cm

Substitute the value of s,a,b,c in the formula above.

Area of Δ = √18(18-9)(18-12)(18-15)

                = √18(9)(6)(3)

                = √18 x 18 x 9

                = √18² x 3²

                = 18 x 3

                = 54 cm²

Now substitute the value of Area of Δ in eq.1

h = 1/9 x ( Area of Δ )

h = 1/9 x 54

h = 6 cm

∴ Height of the parallelogram is 6 cm.

Answer 2

Given sides of the triangle =

9 cm , 12 cm and 15 cm

Finding its area :

Firstly it's semi perimeter =

(9+12+15)/2 = 36/2 = 18 cm

Area of the triangle (Using Heron's Formula) =

$\sqrt{s(s - a)(s - b)(s - c)} \\ \\ = \sqrt{18 \times (18 - 9)(18 - 12)(18 - 15)} \\ \\ = \sqrt{18 \times 9 \times 6 \times 3} \\ \\ = 18 \times 3 = 54 \: {cm}^{2}$

Then it is given in the statement that the area of the parallelogram and the area of triangle are same. Also , the parallelogram is standing on the base of 9 cm .

According to the question :

Base × Height = 54 cm²

9 cm × Height = 54 cm²

Height = 54/9 = 6 cm

The required height of the parallelogram is 6 cm.