Question

The adjacent sides of a parallelogram are 36 cm and 27 cm respectively. if the distance between the shorter side is 12 cm, find the distance between the longer side

Answer 1

Let the parallelogram be ABCD with AB=CD=36 cm and AD=CB=27 cm.
Let DE be the altitude from D on BC and AF be the altitude from A on CD. 

The shortest distance between the shorter sides is 12 cm.

We know that the perpendicular distance is the shortest distance between two points.
So we can say that altitude DE is the shortest distance between the two short sides AD & BC.

Therefore, DE=12 cm.

Now, we know that Area of Parallelogram = ½ x Base x Height. 

So Area of Parallelogram ABCD = ½ × AD × DE = ½ × AF × DC

Area of Parallelogram ABCD = ½ × 27 × 12 = ½ × AF × 36

Area of Parallelogram ABCD = ½ × 324 = ½ × AF × 36       

=  324 =  AF × 36

=  AF   = 324/36

=  AF   =  9 cm 

Hope This Helps :)

Answer 2

Given:

• Two adjacent sides of a parallelogram are 36 cm and 27 cm.
• The distance between longer sides is 12 cm

To find:

• Distance between shorter sides?

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Solution:

$\setlength{\unitlength}{1.6cm}\begin{picture}(8,2)\linethickness{0.4mm}\put(8.6,3){\large\tt{D}}\put(7.7,0.9){\large\tt{A}}\put(9.5,0.7){\sf{\large{27 cm}}}\put(11.1,0.9){\large\tt{B}}\put(8,1){\line(1,0){3}}\qbezier(11,1)(11.5,2)(12,3)\put(9,3){\line(3,0){3}}\put(9.1,1.8){\sf{\large{12 cm}}}\put(9,1){\line(0,1){2}}\qbezier(8,1)(8.5,2)(9,3)\qbezier(11.5,2)(11.5,2)(9,3)\put(12.1,3){\large\tt{C}}\put(11.5,1.6){\sf{\large{36 cm}}}\put(11.65,2){\large\tt{F}}\put(8.9,0.7){\large\tt{E}}\end{picture}$

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Here,

• AB = 27 cm
• BC = 36 cm
• DE = 12 cm

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☯ Let Distance between shorter sides, DF be x cm.

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We know that,

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$\star\;{\boxed{\sf{\purple{Area_{\;(parallelogram)} = Base \times Height}}}}$

Now, Finding area of parallelogram, considering AB as base :

$:\implies\sf AB \times DE$

$:\implies\sf 27 \times 12$

$:\implies{\boxed{\sf{\pink{324\;cm^2}}}}\;\bigstar$

Now, again finding area of parallelogram, considering BC as base :

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$:\implies\sf BC \times DF$

$:\implies\sf 36 \times x$

$:\implies{\boxed{\sf{\pink{36x\;cm^2}}}}\;\bigstar$

⠀⠀⠀━━━━━━━━━━━━━━━━━━━━━━━━━━━━

We know that,

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Area of parallelogram will be the same, if we consider AB as base or BC as base.

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Therefore,

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$:\implies\sf AB \times DE = BC \times DF$

$:\implies\sf 324 = 36x$

$:\implies\sf x = \cancel{ \dfrac{324}{36}}$

$:\implies{\boxed{\sf{\purple{9\;cm}}}}\;\bigstar$

$\therefore\;{\underline{\sf{Distance\;between\;the\;shorter\;sides\;is\; \bf{9\;cm}.}}}$