Question

The length of the sides of trianlge are 7cm,13 cm and 12cm. Find the length of perpendicular from the opposite vertex to the side whose length is 12cm.

Answer 1

Given:-

• First Side = 7cm
• Second Side = 13cm
• Third Side = 12cm

To Find:-

• The length of Perpendicular of the opposite vertex to the side whose length is 12cm.

Formulae used:-

• Area of ∆ = $\rm{\sqrt{ (s) ( s - a ) ( s - b ) ( s - c)}}$

• Area of ∆ = $\rm{\dfrac{1}{2}\times{Base}\times{height}}$

Now,

$\implies\sf{ s = \dfrac{Perimeter}{2}}$

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

$\implies\sf{ s = \dfrac{ 7 + 13 + 12}{2}}$

$\implies\sf{ s = \dfrac{ 32}{2} = 16cm}$

Therefore,

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

$\implies\sf{\sqrt{ (16) ( 16 - 7) ( 16 - 13) ( 16 - 12 )}}$

$\implies\sf{ \sqrt{ (16) (9) (3) (4)}}$

$\implies\sf{\sqrt{1728}}$

$\implies\sf{ 41.56cm^2}$

Now,

$\implies\rm{ Area\:of\:triangle = \dfrac{1}{2}\times{Base}\times{height}}$

$\implies\rm{ 41.56 = \dfrac{1}{2}\times{12}\times{height}}$

$\implies\rm{ 41.56 = 6h}$

$\implies\rm{ h = \dfrac{41.56}{6}}$

$\implies\rm{ h = 6.928cm}$

Hence, The Perpendicular height is 6.928cm.

Answer 2

Given:

• Length of sides of triangle are:

1. 7cm
2. 13cm
3. 12cm

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Need to find:

• The length of perpendicular from the opposite vertex to the side 12cm

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Solution:

${ s = \dfrac{Perimeter}{2}}$

$\implies{ s = \dfrac{ a + b + c}{2}}$

$\implies{ s = \dfrac{ 7 + 13 + 12}{2}}$

$\implies{ s = \dfrac{ 32}{2} = 16cm}$

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We know

$\bold{\red{Area\:of\:triangle = \sqrt{ (s) ( s - a ) ( s - b ) ( s - c)}}}$

$\implies{\sqrt{ (16) ( 16 - 7) ( 16 - 13) ( 16 - 12 )}}$

$=\sqrt{ (16) (9) (3) (4)}$

$=\sqrt{1728}$

$= 41.569cm^2$

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${ Area\:of\:triangle = \dfrac{1}{2}\times{Base}\times{height}}$

$\implies{ 41.569 = \dfrac{1}{2}\times{12}\times{height}}$

$\implies{ 41.569 = 6h}$

$\implies{\red{\large{\bold{\boxed{h = 6.9282cm}}}}}$