Question

Find the area of the triangle whose two sides are 18 cm, 10 cm respectively and perimeter is 42 cm.​

Answer 1

Answer :-

• Area of Triangle = 21√11 cm²

Given :-

• First Side (Side a) = 18 cm
• Second Side (Side b) = 10 cm
• Perimeter of triangle = 42 cm

To Find :-

• Area of Triangle = ?

Explanation :-

In order to find the area of the triangle first we need to find the Third Side (Side c) of the Triangle.

Let the Third Side (Side c) to be 'x' cm.

$\underline { \boxed { \bf \implies Perimeter \: of \: Triangle = Sum \: of \: all \: Sides }}$

$\rm \implies Side \: 1 + Side \: 2 + Side \: 3 = 42 \: cm$

$\rm \implies 18 + 10 + x = 42 \: cm$

$\rm \implies 28 + x = 42 \: cm$

$\rm \implies x = 42 - 28 \: cm$

$\underline { \boxed { \bf \implies x = 14 \: cm }}$

Now, Finding the Area of Triangle using Heron's Formulae.

$\underline { \boxed { \bf \implies Heron's \: Formulae = \sqrt{s (s -a) (s - b) (s-c) } }}$

Where,

• s = Semi - Perimeter of the Triangle
• a = First side of the Triangle
• b = Second side of the Triangle
• a = Third side of the Triangle

$\boxed { \bf \implies Semi \: Perimeter \: of \: Triangle = \frac{a + b + c}{2} }$

$\rm \implies \dfrac{18 + 10 + 14}{2}$

$\rm \implies \dfrac{42}{2}$

$\bf \implies 21 \: cm$

Now Substituting the values,

$\rm \implies \sqrt{s (s -a) (s - b) (s-c) }$

$\rm \implies \sqrt{21 (21 - 18) (21 - 10) (21 - 14) }$

$\rm \implies \sqrt{ 21 \times 3 \times 11 \times 7}$

$\underline { \boxed { \bf \implies 21 \sqrt{11} \: cm^2}}$

Hence, the area of the triangle is 21√11 cm².

@Agamsain

Answer 2

Given thαt,

• The two sides of α triαngle αre 18cm and 10cm.
• Perimeter of the triαngle is 42m.

◾️We need to find the αreα of the triαngle.

_______

Let the unknown side be a.

According to the question,

$\longrightarrow{\sf{a+18+10=42}}$

$\longrightarrow{\sf{a+28=42}}$

$\longrightarrow{\sf{a=42-28}}$

$\longrightarrow{\sf{a=14}}$

Heron's Formulαe - $\displaystyle{\boxed{\sf{\orange{ \sqrt{s(s-a)(s-b)(s-c)}}}}}$

Where, s = semiperimeter and a,b,c = sides.

And

$\displaystyle{\sf{ semiperimeter = \dfrac{a+b+c}{2}}}$

• Substitute the values.

$\implies{\sf{ \dfrac{18+10+14}{2}}}$

$\implies{\sf{ \dfrac{42}{2}}}$

$\implies{\sf{21}}$

• Substitute the values in heron's formulae.

$\displaystyle{\sf{ \sqrt{21(21-18)(21-10)(21-14)}}}$

• Simplify this.

$\longrightarrow{\sf{ \sqrt{21(3)(11)(7)}}}$

• Factorise the numbers.

$\longrightarrow{\sf{ \sqrt{7\times 3\times 3\times 11\times 7}}}$

$\longrightarrow{\sf{ 7\times 3\sqrt{11}}}$

:$\large\implies{\boxed{\sf{\orange{21\sqrt{11}}}}}$

Hence, the αreα of the triαngle is 21√11.

And we αre done! :D

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