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Show that the relation `R` defined on the set `A` of all triangles in a plane as `R={(T_1,\ T_2): T_1` is similar to `T_2)` is an equivalence relation. Consider three right angle triangle `T_1` with sides `3,\ 4,\ 5;` `T_2` with sides `5,\ 12 ,\ 13` and `T_3` with sides 6, 8, 10. Which triangles among `T_1,\ T_2` and `T_3` are related?

Text Solution

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`R={(T_1,\ T_2): T_1` is similar to `T_2)`
R is reflexive since every triangle is similar to itself.
If `(T_1,\ T_2) in R`, then `T_1` is similar to `T_2`.
`T_2` is similar to `T_1`.
R is symmetric.
`(T_1,\ T_2), (T_2,\ T_3) in R`
`T_1` is similar to `T_2` and `T_2` is similar to `T_3`.
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