Question

A triangle and a parallelogram have the same base and 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.With Diagram

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Answer 1

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

To find:

• Height of parallelogram = ?

Solution:

Let ,

length of the sides of triangle be,

• a = 15 cm
• b = 14 cm
• c = 13 cm

Let,

• semi - perimeter of triangle be "s".

Therefore,

${\underline{\boxed{ \bold{ \red{ s = \dfrac{a + b + c}{2}}}}}} \: \: \bigstar$

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

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

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

Now, Using Heron's Formula,

${ \underline{\boxed{\bf{\purple{Area = \sqrt{s(s - a)(s - b)(s - c)}}}}}}$

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

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

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

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

Let, height of parallelogram be "h" cm.

Now, We know that,

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

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

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

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

∴ Thus , Height of Parallelogram is 5.6cm

Answer 2

$\huge\red\dag\mathbb{AnSwEr}\red\dag$

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

To find :

Height of parallelogram = ?

Solution:

Let , length of the sides of triangle be,

a = 15 cm

b = 14 cm

c = 13 cm

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

Therefore,

$\begin{gathered}{\underline{\boxed{ \bold{ \red{ s = \dfrac{a + b + c}{2}}}}}} \: \: \bigstar\\ \\ \end{gathered}$$</p><p>\begin{gathered}:\implies\sf s = \dfrac{15 + 14 + 13}{2}\\ \\\end{gathered}$

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

$\begin{gathered}:\implies\bf \pink{s = 21\;cm}\\ \\\end{gathered}$

Now, Using Heron's Formula,

$\begin{gathered}{ \underline{\boxed{\bf{\purple{Area = \sqrt{s(s - a)(s - b)(s - c)}}}}}}\\ \\\end{gathered}$$</p><p>\begin{gathered}:\implies\sf \sqrt{21(21 - 15)(21 - 14)(21 - 13)}\\ \\\end{gathered}$$\begin{gathered}:\implies\sf \sqrt{21 \times 6 \times 7 \times 8}\\ \\\end{gathered}$$\begin{gathered}:\implies\sf \sqrt{7056}\\ \\\end{gathered}$

$\begin{gathered}:\implies{\boxed{\frak{\pink{84 \:cm^2}}}}\;\\ \\\end{gathered}$

Let, height of parallelogram be "h" cm.

Now, We know that,

$\begin{gathered}{\underline{\boxed{\sf{\purple{Area_{\;(parallelogram)} = Base \times Height}}}}}\\ \\\end{gathered}$

$\begin{gathered}:\implies\sf 15\times h=84\\ \\\end{gathered}$$\begin{gathered}:\implies\sf h =\cancel{\frac{84}{15}}\\ \\\end{gathered}$$\begin{gathered}:\implies{\boxed{\frak{\pink{h=5.6 \:cm}}}}\;\\ \\\end{gathered}$

∴ Thus , Height of Parallelogram is 5.6cm