Question

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

Answer 1

Answer :-

• The height of the parallelogram is 12 cm.

To find :-

• The height of the parallelogram.

Solution :-

• Before finding the height of the parallelogram, we have to find the area of the triangle. We will use Heron's formula for doing so, but for using it, we first have to find the semi-perimeter of the triangle!

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We know that :-

$\underline{\boxed{\sf Semi\!-\!perimeter \: of \: a \: triangle = \dfrac{a + b + c}{2}}}$

Where,

• a = First side.
• b = Second side.
• c = Third side.

Here,

• First side = 26 cm.
• Second side = 28 cm.
• Third side = 30 cm.

Substituting the given values in this formula,

$\hookrightarrow\bf Semi \!-\! perimeter = \dfrac{26 + 28 + 30}{2}$

$\bf \hookrightarrow Semi\!-\!perimeter = \dfrac{84}{2}$

$\hookrightarrow \overline{\boxed{ \bf Semi\!-\!perimeter = 42 \: cm}}$

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• Now let's use the Heron's formula to find the area of the triangle!

Heron's formula states that :-

Area of a triangle is :-

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

Where,

• s = Semi-perimeter.
• a = First side.
• b = Second side.
• c = Third side.

Here,

• Semi-perimeter = 42 cm.
• First side = 26 cm.
• Second side = 28 cm.
• Third side = 30 cm.

Substituting the given values in this formula,

$\rm Area = \sqrt{42 \: (42 - 26) (42 - 28) (42 - 30) \:}$

$\rm Area = \sqrt{42 \: (16) \: (14) \: (12)\:}$

$\rm Area = \sqrt{42 \times 16 \times 14 \times 12 \:}$

$\rm Area = \sqrt{112896\:}$

$\overline{\boxed{\rm Area = 336 \: cm^2}}$

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• Finally, let's find the height of the parallelogram!

It has been given that :-

• The triangle and the parallelogram have the same area.
• The base of the parallelogram is 28 cm.

Which means that :-

• The area of the parallelogram is 336 cm² too.

Now, we know that :-

$\underline{\boxed{\sf Area \: of \: a \: parallelogram = Base \times Height}}$

Here,

• Area = 336 cm².
• Base = 28 cm.

• Let the height be "h" cm.

Substituting the given values in this formula,

$\tt \longmapsto 336 = 28 \times h$

$\tt \longmapsto 336 = 28h$

$\tt \longmapsto \dfrac{336}{28} = h$

$\longmapsto \overline{\boxed{\tt 12 \: cm = h}}$

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• Hence, the height of the parallelogram is 12 cm.