Question

$\large\bold{\underline{\underline{Question:-}}}$


The perimeter of a trapezium is 116m,it's non parallel sides are 16m and 24m and it's altitude is 16m. find the area of the trapezium.​
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Answer 1

$\\\;\underbrace{\underline{\sf{Understanding\;the\;Concept\;:-}}}$

Here the Concept of Perimeter and Areas of Trapezium has been used. We are given the Perimeter of Trapezium and its altitude and non parallel sides. From this we can find a relation between the two parallel sides which are used for calculating its area. After finding this relation, we can use this to find the area of given Trapezium.

Let's do it !!

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★ Formula Used :-

$\\\;\boxed{\sf{Perimeter\;of\;Trapezium\;=\;\bf{Sum\;of\;all\;sides}}}$

$\\\;\boxed{\sf{Area\;of\;Trapezium\;=\;\bf{\dfrac{1}{2}\;\times\;(Sum\;of\;Parallel\;sides)\;\times\;Altitude}}}$

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★ Solution :-

Given,

» Perimeter of Trapezium = 116 m

» First non - parallel side of Trapezium = s₁ = 16 m

» Second non - parallel side of Trapezium = s₂ = 24 m

» Height of Trapezium = Altitude = h = 16 m

• Let the first parallel side of Trapezium be 'a' and the second parallel side of Trapezium be 'b'.

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~ For the relation between a and b :

Let's calculate this using the Perimeter. So,

$\\\;\;\sf{:\rightarrow\;\;Perimeter\;of\;Trapezium\;=\;\bf{Sum\;of\;all\;sides}}$

$\\\;\;\sf{:\rightarrow\;\;Perimeter\;of\;Trapezium\;=\;\bf{s_{1}\;+\;s_{2}\;+\;a\;+\;b}}$

Now by applying the value of Perimeter, we get,

$\\\;\;\sf{:\rightarrow\;\;s_{1}\;+\;s_{2}\;+\;a\;+\;b\;=\;\bf{116}}$

$\\\;\;\sf{:\rightarrow\;\;16\;+\;24\;+\;a\;+\;b\;=\;\bf{116}}$

$\\\;\;\sf{:\rightarrow\;\;40\;+\;a\;+\;b\;=\;\bf{116}}$

$\\\;\;\sf{:\rightarrow\;\;\;a\;+\;b\;=\;\bf{116\;-\;40}}$

$\\\;\;{\bf{:\rightarrow\;\;\;a\;+\;b\;=\;\bf{76}}}$

$\\\;\qquad\quad\large{:\mapsto\;\;\underline{\boxed{\tt{\purple{a\;+\;b\;=\;76}}}}}$

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~ For the Area of Trapezium ::

We already have the value of a + b.

Then,

$\\\;\;\sf{:\Longrightarrow\;\;Area\;of\;Trapezium\;=\;\bf{\dfrac{1}{2}\;\times\;(Sum\;of\;Parallel\;sides)\;\times\;Altitude}}$

$\\\;\;\sf{:\Longrightarrow\;\;Area\;of\;Trapezium\;=\;\bf{\dfrac{1}{2}\;\times\;(a\;+\;b)\;\times\;h}}$

Now let's apply the values we have. Then,

$\\\;\;\sf{:\Longrightarrow\;\;Area\;of\;Trapezium\;=\;\bf{\dfrac{1}{2}\;\times\;(76)\;\times\;16}}$

$\\\;\;\sf{:\Longrightarrow\;\;Area\;of\;Trapezium\;=\;\bf{\dfrac{1}{\cancel{2}}\;\times\;(76)\;\times\;\cancel{16}}}$

$\\\;\;\sf{:\Longrightarrow\;\;Area\;of\;Trapezium\;=\;\bf{76\;\times\;8}}$

$\\\;\;\bf{:\Longrightarrow\;\;Area\;of\;Trapezium\;=\;\bf{608\;\;m^{2}}}$

$\\\;\large{\underline{\underline{\rm{\odot\;Thus,\;area\;of\;Trapezium\;is\;\;\boxed{\bf{\blue{608\;\;m^{2}}}}}}}}$

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★ More Formulas to know :-

$\\\;\tt{\leadsto\;\;Area\;\;of\;\;Rectangle\;=\;Length\;\times\;Breadth}$

$\\\;\tt{\leadsto\;\;Area\;of\;Square\;=\;(Side)^{2}}$

$\\\;\tt{\leadsto\;\;Area\;of\;Circle\;=\;\pi r^{2}}$

$\\\;\tt{\leadsto\;\;Area\;\;of\;\;Triangle\;=\;\dfrac{1}{2}\;\times\;Base\;\times\;Height}$

$\\\;\tt{\leadsto\;\;Area\;\;of\;\;Parallelogram\;=\;Base\;\times\;Height}$

$\\\;\tt{\leadsto\;\;Perimeter\;\;of\;\;Rectangle\;=\;2(Length\;+\;Breadth)}$

$\\\;\tt{\leadsto\;\;Perimeter\;of\;Circle\;=\;2\pi r}$

$\\\;\tt{\leadsto\;\;Perimeter\;of\;Square\;=\;4\;\times\;(Side)}$