Question

A triangle and a parallelogram have the same base and the same area. If the sides of
the triangle are 26 cm, 28 cm and 30 cm, and the parallelogram stands on the base
28 cm, find the height of the parallelogram.​

Answer 1

Given:

• Sides of traingle are 26 cm, 28 cm and 30 cm.
• Base of parallelogram and triangle = 28 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 = 26 cm
• b = 28 cm
• c = 30 cm

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Also, Let semi - perimeter of triangle be s.

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Therefore,

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

$:\implies\sf s = \dfrac{26 + 28 + 30}{2}$

$:\implies\sf s = \cancel{ \dfrac{84}{2}}$

$:\implies\sf \pink{s = 42\;cm}\;\bigstar$

Now, Using Heron's Formula,

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

$:\implies\sf \sqrt{42(42 - 26)(42 - 28)(42 - 30)}$

$:\implies\sf \sqrt{42 \times 16 \times 14 \times 12}$

$:\implies{\boxed{\frak{\pink{336\;cm^2}}}}\;\bigstar$

⠀⠀━━━━━━━━━━━━━━━━━━━━━━━━━━━

☯ Let height of parallelogram be h cm.

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Now, We know that,

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

$:\implies\sf 28 \times h = 336$

$:\implies\sf h = \cancel{ \dfrac{336}{28}}$

$:\implies{\boxed{\frak{\pink{h = 12\;cm}}}}\;\bigstar$

$\therefore\;{\underline{\sf{Thus,\; Height\;of\; Parallelogram\;is\;{\textsf{\textbf{12\;cm}}}.}}}$

Answer 2

Step-by-step explanation:

Given :

• A triangle and a parallelogram have the same base and the same area.

• If the sides of the triangle are 26 cm, 28 cm and 30 cm, .

• the parallelogram stands on the base 28 cm

To Find :

• find the height of the parallelogram.

Solution :

• According to the Question :

Let height be h

$: \implies \: \: \boxed{ \sf \: Area \: of \: parallelogram \: = Base \times Height }$

• Substitute all values

$: \implies \: \: \sf \: Area \: of \: parallelogram \: = 28 \times Height .......Equ( i )$

$: \implies \: \: \boxed{ \sf \: side \: = \frac{a + b + c}{2} }$

• Substitute all values :

$: \implies \: \: \sf \: side \: = \frac{28 + 30+ 26}{2} \\ \\ \\ : \implies \: \: \sf \: side \: = \cancel{\frac{84}{2}} \\ \\ \\ \: : \implies \: \: \sf \: side \: = \: 42$

Find The value of Area of triangle :

$: \implies \: \: \boxed{ \sf \:Area \: of \: triangle = \sqrt{s(s-a)(s-b)(s-c)} }$

• Substitute all values :

$: \implies \: \: \: \: \sf \:Area \: of \: triangle = \sqrt{42(42-28)(42-30)(42-26)} \\ \\ \\ : \implies \: \: \: \: \sf \:Area \: of \: triangle = \sqrt{42 \times 14 \times 8 \times 16} \\ \\ \\: \implies \: \: \: \: \sf \:Area \: of \: triangle = 56 \times 6 ..... Equ ( ii)$

Now we know that area lying in common base in triangle and parallelogram is equal .

Area of parallelogram = Area of triangle

From ( i )  and ( ii )

$: \implies \: \: \: \: \sf \: 28 \times h \: = 56 \times 6 \\ \\ \\ : \implies \: \: \: \: \sf \:h = \frac{ \cancel{56} \times 6}{ \cancel{28}} \\ \\ \\ : \implies \: \: \: \: \sf \:h = 2 \times 6 \\ \\ \\ : \implies \: \: \: \: \sf \:h = 12$

• Hence the  height of the parallelogram is 12 cm .