12. The relation of similarity triangles in the set of triangles is (1) equivalence relation (2) symmetric (3) reflexive (4) transitive
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Step-by-step explanation:
R={(T
1
,T
2
):T
1
is similar toT
2
}
R is reflexive since every triangle is similar to itself.
Further, if (T
1
,T
2
)∈R, then T
1
is similar to T
2
.
⇒T
2
is similar to T
1
.
⇒(T
2
,T
1
)∈R
∴R is symmetric.
Now,
Let (T
1
,T
2
),(T
2
,T
3
)∈R.
⇒T
1
is similar to T
2
and T
2
is similar to T
3
.
⇒T
1
is similar to T
3
.
⇒(T
1
,T
3
)∈R
∴R is transitive.
Thus, R is an equivalence relation.
Now, we can observe that:
6
3
=
8
4
=
10
5
(=
2
1
)
∴ The corresponding sides of triangles T
1
and T
3
are in the same ratio.
Then, triangle T
1
is similar to triangle T
3
.
Hence, T
1
is related to T
3
.
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