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Question
Show that the relation R defined in the set A of all triangles as R = {(T1, T2) : T1 is similar to T2}, is an equivalence relation. Consider three right angle triangles T1 with sides 3, 4, 5, T2 with sides 5, 12, 13 and T3 with sides 6, 8, and 10. Which triangles among T1, T2 and T3 are related?
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Solution
(i) Reflexive:
R = {(T1, T2) : T1 is similar to T2}
R is reflexive since every triangle is similar to itself.
∴ R is reflexive.
(ii) Symmetric:
Further, if (T1, T2) ∈ R, then T1 is similar to T2.
⇒ T2 is similar to T1.
⇒ (T2, T1) ∈ R
∴ R is symmetric.
(iii) Transitive:
Now, let (T1, T2), (T2, T3) ∈ R.
⇒ T1 is similar to T2 and T2 is similar to T3.
⇒ T1 is similar to T3.
⇒ (T1, T3) ∈ R
∴ R is transitive.
Thus, R is an equivalence relation.
Now, we can observe that:
`3/6 = 4/8 = 5/10 (=1/2)`
The corresponding sides of triangles T1 and T3 are in the same ratio.
Then, triangle T1 is similar to triangle T3.
Hence, T1 is related to T3.
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