Question

$\large \; {\underline{\underline{\maltese \; {\purple{\pmb{\sf{ Simple \; Question \; :- }}}}}}}$

$\dashrightarrow$ Find the Area of a triangle if its two sides are 18 cm and 10 cm and its Perimeter is 42 cm .




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$\leadsto$ $\red{\sf{No \; Spams }}$
$\leadsto$ $\green{\sf{Proper \; Explanation \; will \; be \; Appreciated . }}$



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Answer 1

Answer:

$\large\underline{\sf{21 \sqrt{11} cm^{2} \: is \: the \: required \: answer }}$

Step-by-step explanation:

According to question:-

Given:-

• 2 sides of triangle
• Perimeter of triangle.

Also given that:-

• 1st side = 18cm.
• 2nd side =10 cm.
• Perimeter= 42

To find:-

• Area of the triangle.

Solution for the third side:-

We know that sum of sides should be equal to the perimeter.

$\large{\sf{a + b + c = p}}$

Where,

• a = First side of triangle
• b= Second side of triangle
• c = Third side of triangle
• p = perimeter of triangle

$\large\underline{\sf{18cm + 10cm + c = 42cm}}$

$\large\underline{\sf{28cm + c = 42cm}}$

$\large\underline{\sf{c = 42cm - 28cm}}$

$\large\underline{\sf{c = 14cm}}$

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Calculating for area of triangle.

$\large\underline{\sf{ \sqrt{s(s - a)(s - b)(s - c)} }}$

Where,

• s=a+b+c/2
• a=First side of triangle
• b=Second side of triangle
• c= Third side of triangle

Calculating for s:-

$\large \green{\underline{ \red{ \boxed{\sf{ \frac{a + b + c}{2} = s }}}}}$

$\large{\sf{ \implies\frac{18cm + 10cm + 14cm}{2} }}$

$\large{\sf{ \implies \frac{42cm}{2} }}$

$\large\underline{\sf{ \implies 21cm}}$

Calculating for area :-

$\large\underline{\sf{ \implies \sqrt{21(21 - 18)(21 - 10)(21 - 14)} }}$

$\large\underline{\sf{ \implies\sqrt{21(3)(11)(7)} }}$

$\large{\sf{ \implies\sqrt{3 \times 7 \times 3 \times 11 \times 7} }}$

$\large\underline{\sf{ \implies3cm \times 7cm \sqrt{11} }}$

$\large\underline{\sf{ \implies21 \sqrt{11} {cm}^{2} }}$

Hence, The area of the figure is 21√11.

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Answer 2

Given :

• Perimeter = 42 cm
• One Side = 18 cm
• Second Side = 10 cm

To Find :

• Area of the triangle.

Solution :

So, here we are given, perimeter is 42 cm. Also, we are given with the 2 sides. So, firstly we will have to find the 3rd side.

We know that,

$\sf\longrightarrow\;Perimeter_{(Triangle)} = Sum\; of\; all\; sides$

Let the missing side be x.

Putting the values,

$\sf\longrightarrow\;Perimeter_{(Triangle)} = Sum\; of\; all\; sides$

$\sf\longrightarrow\;42 = 18 \ + 10 + x$

$\sf\longrightarrow\;42 =28+ x$

$\bf\longrightarrow\;14 \: cm= x$

So, the other side is 14 cm.

Now, finding the area of the triangle.

★ Firstly, finding the semi perimeter :

$\longrightarrow \sf \dfrac{a + b + c}{2}$

Substituting the values,

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

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

$\longrightarrow \sf \cancel \dfrac{42}{2}$

$\longrightarrow \: \underline{ \bf 21 \: cm}$

★ Using Heron's Formula :

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

[Here, s stands for semi - perimeter and a, b and c are the 3 sides.]

Substituting the values,

$\sf\longrightarrow\; Area = \sqrt{21(21 - 18) (21 - 10) (21 - 14) }$

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

$\sf\longrightarrow\; Area =\sqrt{21 \times 3 \times 11 \times 7 }$

$\sf\longrightarrow\; Area =\sqrt{4851 }$

$\sf\longrightarrow\; Area = 69.6491$

$\sf\longrightarrow\; Area \approx \: 69.65 \: {cm}^{2}$

$\color{hotpink}{\sf\longrightarrow\; \underline{\underline{\boxed{ \bf Area = 69.65 \: {cm}^{2} }} }\: \bigstar}$

Hence, the area of the triangle is 69.65 cm².