Question

sides of triangle are in the ratio 5:12:13 and its perimeter is 360 cm. finds the area of triangle

Answer 1

Answer:

4320 cm²

Step-by-step explanation:

As per the provided information in the given question, we have :

• Sides of triangle are in the ratio 5:12:13.
• Perimeter of the triangle is 360 cm

We've been asked to calculate area of triangle. In order to calculate the area of the triangle, firstly we need to calculate the sides.

⠀⠀⠀⠀⠀» Let us assume the sides as 5x, 12x and 13x respectively. Now, according to the question the perimeter of the triangle is 360 cm. The perimeter of the triangle is the sum of its all sides. So, writing this statement in the form of a linear equation.

$\dashrightarrow \quad \rm {P_{(\Delta)} = Sum_{(All \; sides)} }$

Substitute the expressions of its sides we have assumed. Also, substitute the value of perimeter to make it a linear equation.

$\dashrightarrow \quad \rm {360 = 5x + 12x + 13x }$

Performing the addition of the like terms in the RHS.

$\dashrightarrow \quad \rm {360 = 30x }$

Now, transposing 30 from RHS to LHS in order to perform division to find the value of x.

$\dashrightarrow \quad \rm {\cancel{\dfrac{360}{30}} = x }$

$\dashrightarrow \quad\underline{\boxed{ \bf {12 = x }}}$

Therefore the value of x is 12.⠀⠀⠀

$\rule{200}2$

$\large {\underline { \sf {Sides \; Of \; Triangle :}}}$

• First side = 5x = 5(12) cm = 60 cm
• Second side = 12x = 12(12) cm = 144 cm
• Third side = 13x = 13(12) cm = 156 cm

$\rule{200}2$

Now, we'll be calculating the area of the triangle. In order to calculate the area, we also need the semi-perimeter of the triangle.

⠀⠀⠀⠀⠀» Basically, semi-perimeter means half of the perimeter in the simple words. Here we already have perimeter that is 360 cm. So, its half will be the semi-perimeter of the triangle. To find the half of anything we divide it by 2. When 360 cm is divided by 2, we get 180 cm. So, the semi-perimeter of the triangle is 180 cm.

Now, as we know that,

$\bigstar \quad \underline{\boxed{ \bf {Area_{(\Delta)} = \sqrt{s(s-a)(s-b)(s-c)} }}}$

Here,

• a , b , c are sides of triangle.
• s is semi-perimeter.

Now, substitute the value to get the required answer! (Also, denoting Area as A.)

$\dashrightarrow \quad \rm {A = \sqrt{180(180-60)(180-144)(180-156)} \; cm^2 }$

Performing multiplication in the brackets.

$\dashrightarrow \quad \rm {A = \sqrt{180(120)(36)(24)} \; cm^2 }$

Multiplying the terms under the square roots.

$\dashrightarrow \quad \rm {A = \sqrt{18662400} \; cm^2 }$

Writing the square root of 18662400 which will be the area of the triangle.

$\dashrightarrow \quad \underline{\boxed {\bf {A = 4320 \; cm^2 } }}$

∴ The area of the triangle is 4320 cm².

Answer 2

$\purple{ \huge \underline{ \boxed{ \mathbb \red{ \colorbox{cyan}{QUESTION :} }}}}$

Sides of triangle are in the ratio 5:12:13 and its perimeter is 360 cm. Find the area of triangle .

$\purple{ \huge \underline{\boxed{\mathbb \red{\colorbox{cyan}{ANSWER:} }}}}$

$\green{\huge\underline{\underline{ \bf Given : }}}$

• Sides of triangle are in the ratio 5 : 12 : 13
• Perimeter of triangle = 360 cm

$\green{\huge\underline{\underline{ \bf To \: find : }}}$

• Area of the triangle

$\green{\huge\underline{\underline{ \bf Formuals \: used: }}}$

$\purple{\boxed{\bf Perimeter_ {(\triangle)} = Sum \: of \: all \: sides}}$

Pythagoras Theorem

$\purple{\boxed{\bf {H}^{2} = {B}^{2} + {P}^{2} }}$

$\purple{\boxed{\bf Area \: of \: right \: angle \: triangle = \frac{1}{2} \times b \times h }}$

$\green{\huge\underline{\underline{ \bf Solution: }}}$

Let us assume the sides as 5x , 12x and 13x .

$\bf Perimeter_ {(\triangle)} = 5x + 12x + 13x \\ \\ \bf \implies 360 = 30x \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \bf \implies 30x = 360 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \bf \implies x = \cancel \frac{360}{30} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \bf \implies x = 12 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

$\blue{\underline{ \underline{\bf Sides \: of \: Triangle: }}}$

$\bf \longrightarrow 5x = 5 \times 12 = 60 \: cm \: \: \: \: \: \: \: \: \\ \\ \bf\longrightarrow12x = 12 \times 12 = 144 \: cm \\ \\ \bf\longrightarrow13x = 13 \times 12 = 156 \: cm$

Let us check that this triangle is a right angle triangle or not .

We know that hypotenuse is longest side in triangle . Therefore , Let H = 156 cm then B = 144 cm and P = 60 cm where B and P are interchangeable .

$\bf \longrightarrow {(156)}^{2} = {(144)}^{2} + {(60)}^{2} \\ \\\bf \longrightarrow24336 = 20736 + 3600 \\ \\ \bf \longrightarrow24336 = 24336 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

Therefore , it is a right angle triangle .

Base ( B ) = 144 cm

Height ( H ) = Perpendicular = 60 cm

$\bf Area_{( \triangle)}= \frac{1}{ \cancel2} \times 144 \times \cancel{60} \\ \\\bf Area_{( \triangle)} = 144 \times 30 \: \: \: \: \: \: \: \: \\ \\ \bf Area_{( \triangle)} = 4320 \: {cm}^{2} \: \: \: \: \: \: \\ \\ \purple{ \boxed{\bf Area_{( \triangle)} = 4320 \: {cm}^{2} }} \: \:$