Question

A triangle and a parallelogram have the same base and the same area. If the sides of

the triangle are 26 cm. 28 cm and 30 cm. And the parallelogram stands on the base

28 cm, find the height of the parallelogram.



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Answer 1

Firstly find the area of a triangle by heron’s formula and area of parallelogram then equate their areas to calculate the height of a parallelogram.

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Solution:

Let the Length of the sides of the triangle are a=26 cm, b=28 cm and c=30 cm.

Let s be the semi perimeter of the triangle.

s=(a+b+c)/2

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

s = 42 cm

Using heron’s formula,

Area of the triangle = √s (s-a) (s-b) (s-c)

= √42(42 – 26) (46 – 28) (46 – 30)

= √42 × 16 × 14 × 12

=√7×6×16×2×7×6×2

√7×7×6×6×16×2×2

7×6×4×2= 336 cm²

Let height of parallelogram be h.& Base= 28 (given)

Area of parallelogram = Area of triangle (given)

[Area of parallelogram =base× height]

28× h = 336

h = 336/28 cm

h = 12 cm

Hence,

The height of the parallelogram is 12 cm.

Answer 2

$\large\bf\underline{Question:}$

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

$\large\bf\underline{Solution:}$

Let the side of the triangle be

a = 26cm

b = 28cm

c = 30cm

First,we will find the area of the triangle using Heron's Formula.

For that we need to find semi-perimeter.

$s = \frac{a + b + c}{2} \\ s = \frac{26 + 28 + 30}{2} \\ s = \frac{84}{2} \\ s = 42cm$

Heron's Formula:

$\boxed{\sf{Area = \sqrt{s(s - a)(s - b)(s - c)}}}$

$Area = \sqrt{42(42 - 26)(42 - 28)(42 - 30)} \\ Area = \sqrt{42 \times 16 \times 14 \times 12} \\ Area = \sqrt{ 6 \times 7</u><u>\</u><u>t</u><u>i</u><u>m</u><u>e</u><u>s</u><u> </u><u>1</u><u>6</u><u> \times 7 \times 2 \times 6 \times 2 } \\ Area = 6 \times</u><u> </u><u>4</u><u>\</u><u>t</u><u>i</u><u>m</u><u>e</u><u>s</u><u> </u><u>7 \times 2 \\ Area = </u><u>33</u><u>6</u><u>c {m}^{2}$

Its given that the area of parallelogram is equal to the area of triangle.

$Area \: of \: parallelogram=Area \: of \: triangle \\ base \times height = 336 \\ 28 \times height = 336 \\height = 12cm$

Hence, the height of the parallelogram is 3cm.

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