Question

A triangle and a parallelogram have the same base
and the same area. If the sides of the triangle are
15 cm, 14 cm and 13 cm and the parallelogram
stands on the base 15 cm, find the height of
parallelogram.​

Answer 1

✬ Height = 5.6 cm ✬

Step-by-step explanation:

Given:

• A triangle and parallelogram have same area.
• Sides of triangle are 15 , 14 and 13 cm.
• Measure of base of parallelogram is 15 cm.

To Find:

• What is the height of parallelogram ?

Solution: Let ABC be a triangle where,

• a = 13 cm , b = 14 cm , c = 15 cm

We have to find the area of triangle by using Heron's formula.

➟ Semi Perimeter (S) = ( a + b + c/2 )

➟ Semi Perimeter = ( 13 + 14 + 15/2 )

➟ S = 42/2 = 21

★ Heron's Formula ∆ = √s (s – a) (s – b) (s – c) ★

$\implies{\rm }$ Area of ∆ = √21 (21 – 13) (21 – 14) (21 – 15) cm²

$\implies{\rm }$ √21 $\times$ 8 $\times$ 7 $\times$ 6 cm²

$\implies{\rm }$ √7056 cm²

$\implies{\rm }$ √84 $\times$ 84 cm²

$\implies{\rm }$ 84 cm²...........(1)

So, The area of triangle is 84 cm².

Now, Finding the area of parallelogram. As we know that :-

★ Area of Parallelogram = ( Base $\times$ Height ) ★

$\implies{\rm }$ Area of ||gm = ( 15 $\times$ Height ) cm²

$\implies{\rm }$ 15h cm²........(2)

According to the question,

➭ Equation 1 = Equation 2

$\implies{\rm }$ 84 = 15h

$\implies{\rm }$ 84/15 = Height

$\implies{\rm }$ 5.6 cm = Height

Hence, the height of the parallelogram is 5.6 cm.

Answer 2

GIVEN:

• A triangle and a parallelogram have the same base and the same area.
• the sides of the triangle are 15 cm, 14 cm and 13 cm
• The base of a parallelogram is 15 cm

TO FIND:

• What is the height of the parallelogram ?

SOLUTION:

Let the height of the parallelogram be 'h' cm

We have to find the area of the triangle using Heron's formula

• a = 15 cm
• b = 14 cm
• c = 13 cm

Firstly, we have to find the semi-perimeter of the triangle

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

$\rm{ s = \dfrac{15+ 14+13}{2}}$

$\rm{ s = \cancel\dfrac{42}{2}}$

$\bf{s = 21 \: cm}$

We know that the formula for finding the area of triangle is:-

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

Put the values in the formula

$\rm{\mapsto Area = \sqrt{21(21-15)(21-14)(21-13)}}$

$\rm{\mapsto Area = \sqrt{21 \times 6 \times 7 \times 8}}$

$\rm{\mapsto Area = \sqrt{ 7056}}$

$\bf{\mapsto Area = 84 \: cm^2}$

❝ The area of triangle is 84 cm² ❞

We have given that, the triangle and the parallelogram stands on the same base and have equal areas

✯ Area of triangle = Area of Parallelogram ✯

We know that the formula for finding the area of parallelogram is:-

$\bf{ Area = Base \times Height}$

According to question:-

$\rm{\mapsto 84 = 15 \times h}$

$\rm{\mapsto h = \cancel\dfrac{84}{15}}$

$\bf{\mapsto h = 5.6 \: cm}$

❝ Hence, the height of the parallelogram is 5.6 cm ❞

____________________

VERIFICATION:

➠ $\bf{ 84 = 15 \times 5.6}$

➠ $\bf{ 84 = 84}$

$\rm{[L.H.S. = R.H.S.]}$

VERIFIED....