Question

The perimeter of two similar triangles are 30 cm and 20 cm respectively. If one side of the first triangle is 12cm, determine the corresponding side of the second triangle.

Answer 1

Hint: In this particular question use the concept that perimeter of any shape is the sum of all the sides length, and use the concept that in similar triangles the ratio of their corresponding sides are equal so, use these concepts to reach the solution of the question.

Complete step-by-step answer:

Let ABC and abc are two similar triangles as shown in the above figure.
The sides of the triangle ABC are x, y and z cm respectively.
And the sides of the triangle abc are p, q and r cm respectively.
Now it is given that one of the sides of the triangle ABC is 12 cm.
Let, x = 12cm as shown in the above figure.
So we have to find the corresponding side of the similar triangle.
I.e. P =?
Now it is given that the perimeter of triangles are 30 and 20 cm respectively.
So, ${P_{ABC}} = 30$cm, and ${P_{abc}} = 20$cm.
Now as we know that the perimeter is the sum of all the side lengths.
So, 30 = x + y + z............... (1)
And 20 = p + q + r................ (2)
Now as we know that in similar triangles the ratio of their corresponding sides are equal.
\[ \Rightarrow \dfrac{x}{p} = \dfrac{y}{q} = \dfrac{z}{r}\]
Let the ratio is equal to the constant value K, so we have,
\[ \Rightarrow \dfrac{x}{p} = \dfrac{y}{q} = \dfrac{z}{r} = K\]............... (3)
$ \Rightarrow x = Kp$, $y = Kq$, and $z = Kr$
Now add these three equation we have,
$ \Rightarrow x + y + z = K\left( {p + q + r} \right)$
Now from equation (1) and (2) we have,
$ \Rightarrow 30 = K\left( {20} \right)$
$ \Rightarrow K = \dfrac{3}{2}$
Now from equation (3) we have,
\[ \Rightarrow \dfrac{x}{p} = \dfrac{y}{q} = \dfrac{z}{r} = \dfrac{3}{2}\]
\[ \Rightarrow \dfrac{{12}}{p} = \dfrac{y}{q} = \dfrac{z}{r} = \dfrac{3}{2}\]
\[ \Rightarrow \dfrac{{12}}{p} = \dfrac{3}{2}\]
\[ \Rightarrow p = 12 \times \dfrac{2}{3} = 8\] cm.
So the corresponding side of the similar triangle is 8cm.
So this is the required answer.

Note: Whenever we face such types of questions the key concept we have to remember is that always recall the definition of perimeter of any shape which is stated above, and also recall the properties of similar triangle, so first apply the property and write the values of sides of one triangle in terms of the corresponding sides of the similar triangle as above, then add them as above and substitute the values of perimeter and again simplify we will get the required answer.