Question

the ratio of base to height of a parallelogram is 2:3 if the area of the parallelogram is 864cm^2 then find its base and height​

Answer 1

DIAGRAM :

$\setlength{\unitlength}{1 cm}\begin{picture}(20,15)\thicklines\qbezier(1,1)(1,1)(6,1)\qbezier(1,1)(1,1)(2,4)\qbezier(2,4)(1.6,4)(7,4)\qbezier(6,1)(6,1)(7,4)\qbezier(2,4)(2,4)(2,1)\qbezier(2,4)(1.6,4)(6.5,2.5)\put(0.7,0.5){\sf B}\put(6,0.5){\sf C}\put(1.8,4.3){\sf A}\put(7,4.3){\sf D}\put(2,0.5){\sf N}\put(3.3,0.5){\sf 2k cm}\put(6.7,1.5)\put(6.8,2.4){\sf M}\put(2.2,2.2){\sf 3k}\end{picture}$

$\bold{Given}\begin{cases} \sf \:T he \: ratio \: of \: base \: to \: height \: of \: a \: paralleogram \: is \: \bold{2 :3 }. \\ \\ \sf \: Area \: of \: the \: parallelogram \: is \: \bold{864 \: {cm}^{2} }.\end{cases}$

Need to find : Base and height of the parallelogram.

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Put k in the base to height ratio of the parallelogram.

$\: \:$

• i) Base of the parallelogram = 2k
• ii) Height of the parallelogram = 3k

$\: \:$

For finding the area of a parallelogram, formula is given as :

$\: \:$

$\star \: \underline{\boxed{\sf\purple{Area_{(parallelogram)} = base \: \times height}}}$

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$★ \: \mathfrak {\underline{substitutin g\: the \: values \: : }}$

$\:$

$\sf:\implies \: Area_{(parallelogram)} = AN \times BC \\ \\ \\ \sf:\implies \:864 = 3k \times 2k \\ \\ \\ \sf:\implies \:864 = 6 {k}^{2} \\ \\ \\ \sf:\implies \:k {}^{2} = \cancel\frac{864}{6} \\ \\ \\ \sf:\implies \: {k}^{2} = 144 \\ \\ \\ \sf:\implies \: \underline{\boxed{\mathfrak\purple{k = 12}}} \: \bigstar \\ \\ \\ \\ \sf\therefore{\underline{Hence, \: the \: value \: of \: k \: is \: \bold{12}.}}$

⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀____________________

$\: \:$

Now, putting the value of k in the ratio.

$\: \:$

• i) Base of the parallelogram = 2k = 2(12) = 24cm
• ii) Height of the parallelogram = 3k = 3(12) = 36cm

$\: \:$

$\sf\therefore{\underline{Hence, \: the \: base \: and \: height \: of \: the \: parallelogram \: are \: \bold{24\: cm} \: and \: \bold{36 \: cm} \: respectively. }}$