Question

The side of a triangle are in the ratio 13:14:15 and its perimeter is 84cm. Find the area of triangle sides.

Answer 1

$\bf{\huge{\underline{\boxed{\sf{\blue{Answer:}}}}}}$

$\bf{\underline{\underline{\tt{\red{Given:}}}}}$

• The sides of a triangle are in the ratio 13:14:15
• Perimeter of the triangle is 84 cm

$\bf{\underline{\underline{\tt{\red{To\:find:}}}}}$

• Area of the triangle

$\bf{\underline{\underline{\tt{\red{Solution:}}}}}$

Let x be the common multiple for the ratio of the sides of the triangle.

•°• First side = a = 13x

Second side = b = 14x

Third side = c = 15x

Using the formula for perimeter of a triangle, we can further solve the question and thereby derive the value of x.

$\bf{\large{\underline{\boxed{\sf{\orange{Perimeter\:of\:a\:triangle\:=\:a\:+\:b\:+\:c\:}}}}}}$

Plug in the values,

=> 84 = 13x + 14x + 15x

=> 84 = 27x + 15x

=> 84 = 42x

=> $\large\frac{84}{42}$ = x

=> x = 2

Substitute x = 2 in the values of the ratio of sides of the triangle.

$\bf{\large{\underline{\boxed{\sf{\pink{First\:side\:=\:a=\:13x\:=\:13\times\:2=\:26\:cm}}}}}}$

$\bf{\large{\underline{\boxed{\sf{\pink{Second\:side\:=\:b=\:14x\:=\:14\times\:2=\:28\:cm}}}}}}$

$\bf{\large{\underline{\boxed{\sf{\pink{Third\:side\:=\:c=\:15x\:=\:1</p><p>5\times\:2=\:30\:cm}}}}}}$

Now, to find the area of the triangle, we will use herons formula. For this we need to find the semiperimeter "s"

=> Semiperimeter,s = $\large\frac{Perimeter}{2}$

=> Semiperimeter,s = $\large\frac{84}{2}$

=> Semiperimeter, s = 42

Let's move to heron's formula,

=> Area = $\sqrt{s(s-a) (s-b) (s-c)}$

Substitute the appropriate values of s, a, b and c in the formula.

=$\sqrt{42(42-26) (42-28) (42-30)}$

= $\sqrt{42(16) (14) (12)}$

= $\sqrt{42\times\:224\times\:12}$

= $\sqrt{42\times\:2688}$

= $\sqrt{112896}$

=> Area = 336 sq.cm

$\bf{\large{\underline{\boxed{\sf{\blue{Area\:of\:triangle\:=\:336\:cm^2}}}}}}$

Answer 2

a = 13x

b = 14x

c = 15x

13x + 14x + 15x = 84

42x = 84

x = 84/42

x = 2

a = 26

b = 28

c = 30

s = Perimeter/2 = 84/2 = 42

Area = √s (s-a) (s-b) (s-c)

Area = √ 42 (42-26)(42-28)(42-30)

Area = √42 (16)(14)(12)

Area = √42 × 2688

Area = √ 112896

Area = 336 cm²