Question

the perimeter of a triangle is 36 cm if the altitudes of the triangle are in the ratio 1:2:3 then find length of the corrcorresponding sides of the triangle ​

Answer 1

Answer:

x + 2x + 3x = 36

6x = 36

x = 6

Sides of the triangle are 6 cm, 12 cm, 18 cm

Answer 2

Step-by-step explanation:

Let the sides of the triangle be a, b and c

Let $\rm h_{1} , \: h_{2} \: and \: h_{3}$ be the altitudes of the triangle.

Given, the altitudes of the triangle are in the ratio 1 : 2 : 3

So Let,

• $\rm h_{1} = h$

• $\rm h_{2} = 2h$

• $\rm h_{3} = 3h$

As , Area of the triangle = ½ × base × height

Area of triangle :-

$\rm \dashrightarrow \dfrac{1}{2} \times a \times h_{1} = \dfrac{1}{2} \times b \times h_{2} = \dfrac{1}{2}\times c \times h_{3}$

$\rm \dashrightarrow a = 2b = 3c$

So:-

$\rm \dashrightarrow a = 2b \: , \: a = 3c$

Given, perimeter of the triangle = 36cm

$\rm \dashrightarrow a + b + c = 36$

$\rm \dashrightarrow a + \dfrac{a}{2} + \dfrac{a}{3} = 36$

$\rm \dashrightarrow \dfrac{6a + 3a + 2a }{6} = 36$

$\rm \dashrightarrow \dfrac{11a }{6} = 36$

$\rm \dashrightarrow a = 36 \times \dfrac{6 }{11}$

$\rm \dashrightarrow a = \dfrac{216 }{11}$

Now, let us find the sides of the triangle

$\dashrightarrow a = \dfrac{216}{11}$

$\dashrightarrow b = \dfrac{a}{2} = \dfrac{108}{11}$

$\dashrightarrow c = \dfrac{a}{3} = \dfrac{73}{11}$

$\underline{\boxed{\rm \therefore The \: sides \: of \: the \: triangle \: are \: \dfrac{216}{11}cm\: , \dfrac{108}{11}cm \: and \: \dfrac{72}{11}cm}}$