This page contains the NCERT mathematics class 12 chapter Relations and Functions Exercise 1.1 Problem 12 Solution. Solutions for other problems are available at Exercise 1.1 Solutions
Exercise 1.1 Problem 12 Solution
12. Show that the relation R defined in the set A of all triangles as R = {(T1, T2) : T1 is similar to T1}, is equivalence relation. Consider three right angle triangles T1 with sides 3, 4, 5, T2 with sides 5, 12, 13 and T1 with sides 6, 8, 10. Which triangles among T1, T2 and T3 are related?
To Check whether R is Reflexive: The relation R in the set A is reflexive if (a, a) ∈ R, for every a ∈ A.
We know that every triangle T is similar to itself.
⇒ (T, T) ∈ R
∴ R is reflexive.
To Check whether R is Symmetric: The relation R in the set A is symmetric if (a_1, a_2) ∈ R implies that (a_2, a_1) ∈ R, for all a_1, a_2 ∈ A
If a triangle T1 is similar to another triangle T2
⇒ Triangle T2 is also similar to the triangle T1
⇒ If (T1, T2) ∈ R then (T2, T1) ∈ R
∴ R is symmetric.
To Check whether R is Transitive: The relation R in the set A is transitive if (a_1, a_2) ∈ R and (a_2, a_3) ∈ R implies that (a_1, a_3) ∈ R, for all a_1, a_2, a_3 ∈ A
If there are 3 triangles such that triangle T1 is similar to triangle T2 and triangle T2 is similar to triangle T3 then triangle T1 is similar to triangle T3
⇒ If (T1, T2) ∈ R and (T2, T3) ∈ R then (T1, T3) ∈ R
∴ R is transitive.
∴ R is reflexive, symmetric and transive. Hence, R is an equivalence relation.
Consider three right angle triangles T1 with sides 3, 4, 5, T2 with sides 5, 12, 13 and T1 with sides 6, 8, 10. Which triangles among T1, T2 and T3 are related?
The relation R is defined as R = {(T1, T2): T1 is similar to T2}
Triangle T1 has the sides 3, 4, 5
Triangle T2 has the sides 5, 12, 13
Triangle T3 has the sides 6, 8, 10
As per the definition, any two triangles are similar if the ratio of their corresponding sides is same.
Obviously, if we consider the triangles T1 and T3, the ratio of their sides is
{\dfrac{3}{6} = \dfrac{4}{8} = \dfrac{5}{10} = \dfrac{1}{2}}
So, the ratio of the corresponding sides is same.
⇒ The triangles T1 and T3 are similar triangles.
⇒ Among the given 3 triangles, the triangles T1 and T3 are related.
⇒ (T1, T3) ∈ R
