Question

Q5.A triangle and a parallelogram are on the same base and have the same area. If the sides of the triangle are 15 cm, 14 cm and 13 cm and the parallelogram stands on the base 15 cm, find the height of the parallelogram.

Answer 1

the height of the parallelogram is 5.6 cm

Answer 2

Given:

• Sides of traingle are 15 cm, 14 cm and 13 cm.
• Base of parallelogram and triangle = 15 cm
• Area of triangle = Area of Parallelogram

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To find:

• Height of parallelogram?

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Solution:

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☯ Let length of the sides of triangle be,

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• a = 15 cm
• b = 14 cm
• c = 13 cm

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Also, Let semi - perimeter of triangle be "s".

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Therefore,

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$\star\;\sf s = \dfrac{a + b + c}{2}$

$:\implies\sf s = \dfrac{15 + 14 + 13}{2}$

$:\implies\sf s = \cancel{ \dfrac{42}{2}}$

$:\implies\sf \pink{s = 21\;cm}\;\bigstar$

Now, Using Heron's Formula,

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$\star\;{\boxed{\sf{\purple{Area = \sqrt{s(s - a)(s - b)(s - c)}}}}}$

$:\implies\sf \sqrt{21(21 - 15)(21 - 14)(21 - 13)}$

$:\implies\sf \sqrt{21 \times 6 \times 7 \times 8}$

$:\implies\sf \sqrt{7056}$

$:\implies{\boxed{\frak{\pink{84\;cm^2}}}}\;\bigstar$

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☯ Let height of parallelogram be "h" cm.

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Now, We know that,

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

$:\implies\sf 15 \times h = 84$

$:\implies\sf h = \cancel{ \dfrac{84}{15}}$

$:\implies{\boxed{\frak{\pink{h = 5.6\;cm}}}}\;\bigstar$

$\therefore\;{\underline{\sf{Thus,\; Height\;of\; Parallelogram\;is\;{\textsf{\textbf{5.6\;cm}}}.}}}$