Question

The adjacent sides of a parallelogram ABCD measure 34 cm and 20 cm, and the diagonal
AC measures 42 cm. Find the area of the parallelogram.
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Answer 1

★Given:-

• ABCD is a parallelogram
• Adjacent sides are 34cm & 20cm
• Diagonal = 42cm

★To find:-

• Area of parallelogram.

★Solution:-

The diagonal divides the parallelogram into two congruent triangles.

We have:

• a = 20cm
• b = 34cm
• c = 42cm

Using the formula,

$\mapsto \underline{\boxed{\sf s = \dfrac{a+b+c}{2} }}$

Putting values,

$\displaystyle :\implies \sf s =\frac{20+34+42}{2} \\\\:\implies \sf s = 48cm$

Using the formula,

$\displaystyle \mapsto \underline{\boxed{\sf Area.\triangle = \sqrt{s(s-a)(s-b)(s-c)}}}$

Putting values in the formula,

$\displaystyle :\implies \sf Ar.\triangle ACD = \sqrt{s(s-a)(s-b)(s-c)} \\\\:\implies \sf Area = \sqrt{48(48-20)(48-34)(48-42)} \\\\:\implies \sf Area = 336cm^2$

As the diagonal divides the parallelogram into two congruent triangles,

→Ar.ΔACD = Ar.ΔBAC

Therefore,

⇒Ar.ABCD = 2(336)

                   = 672cm²

Hence,

The area of parallelogram is  672cm²

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Answer 2

$\sf {\large{ \underbrace{ \underline{Understanding \: the \: Question}}}}$

Here we have to find the area of parallelogram. As we know that diagonal of a parallelogram divides it into two equal area or we can say congruent triangles.We can find the area of parallelogram by finding the area of triangles separately.

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GIVEN:

➢ABCD is parallologram

➢AB=34cm

➢BC=20cm

➢AC=42cm[Diagonal]

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TO FIND:

➢Area of parallelogram ABCD

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SOLUTION:

In triangle ABC

By Applying “HERON'S FORMULA”

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Sides of triangle are:-

• a=34cm
• b=20cm
• c=42cm

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$\mapsto\: { \boxed{ \sf{ S= \dfrac{a+b+c}{2}}}}$

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$\sf{:\implies S = \dfrac{34 + 20 + 42}{2} }$

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$\sf{:\implies S = \dfrac{96}{2} }$

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$\mapsto \large{\boxed{ { \sf{ \blue{:\implies S=48}}}}}$

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Now we have formula

$\mapsto \begin{gathered} {\boxed{ \sf{Area \: of \: triangle= \sqrt{S(S-a)(S-b)(S-c)} }}}\\\end{gathered}$

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$\sf{ ar.(ABC) = \sqrt{48(48 - 34)(48 -20)(48 - 42 } }$

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$\sf{ ar.(ABC) = \sqrt{48(14)(28 )(6) } }$

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$\sf{ar.(ABC)=336cm^{2}}$

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Hence the area of triangle ABC is 336cm².

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Now triangle ABC and triangle ADC are congruent.

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It means

• ar.(ABC)=ar.(ACD)

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➛ar.(ACD)+ar.(ABC)=ar.(ABCD)

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➛336cm²+336cm²=672cm²

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Hence the area of parallelogram is 672cm²

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$\sf {\large{ \underbrace{ \underline{More \: Formula \: to \: know}}}}$

》Area of rectangle=L×B

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》Area of square=Side²

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》Area of ∆=½×Base×Height

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》Area of circle=πr²