Home
Class 12
MATHS
Show that the relation R defined in the...

Show that the relation R defined in the set A of all triangles as `R={(T_1,T_2): T_1( i s s i m i l a r t o T)_2}`, is equivalence relation. Consider three right angle triangles `T_1`with sides 3, 4, 5, `T_2`with sides 5, 12, 13 and `T_3`w

Text Solution

Verified by Experts

Here A = set of all triangles
and R =`{(T_1, T_2) : T_1` is similar to `T_2`}
`because ` Every triangle is similar to itself.
`therefore R` is reflexive.
Let `T_1, T_2 in A and (T_1, T_2 ) in R`
`therefore R` is symmetric.
Let `" "T_1, T_2, T_3 in A`
and `" "(T_1, T_2) in R and (T_2, T_3) in R`
`rArr T_1` is similar to `T_2 and T_2` is similar to `T_3`.
`rArr T_1` is similar to `T_3`.
`rArr (T_1, T_3) in R`
`therefore R` is tansitive.
`because R` is reflexive, symmetric and transitive.
`therefore ` R is an equivalence relation.
Now `" " (3)/(6) = (4)/(8) = (5)/(10) = (1)/(2)`
`rArr` The corresponding sides of `T_1 and T_3` are proportional.
`therefore T_1 and T_3` are similar.
`rArr ` Triangle `T_1` is related to triangle `T_3`.
Promotional Banner

Topper's Solved these Questions

  • RELATIONS AND FUNCTIONS

    NAGEEN PRAKASHAN ENGLISH|Exercise Exercise 1.2|12 Videos

  • RELATIONS AND FUNCTIONS

    NAGEEN PRAKASHAN ENGLISH|Exercise Exercise 1.3|14 Videos

  • RELATIONS AND FUNCTIONS

    NAGEEN PRAKASHAN ENGLISH|Exercise Exercies 1e|10 Videos

  • PROBABIILITY

    NAGEEN PRAKASHAN ENGLISH|Exercise Miscellaneous Exercise|19 Videos

  • THREE-DIMENSIONAL GEOMETRY

    NAGEEN PRAKASHAN ENGLISH|Exercise Miscellaneous Exercise|23 Videos

Similar Questions

Explore conceptually related problems

Show that the relation R defined in the set A of all triangles as R={(T_(1),T_(2)):T_(1) is similar to T_(2) }, is equivalence relation.

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?

Show that the relation R defined in the set A of all polygons as R={(P_1,P_2):P_1" and "P_2" have same number of sides"} , is an equivalence relation. What is the set of all elements in A related to the right angle triangle T with sides 3,4 and 5?

Show that the relation R , defined on the set A of all polygons as R={(P_1,\ P_2): P_1 and P_2 have same number of sides}, is an equivalence relation. What is the set of all elements in A related to the right angle triangle T with sides 3, 4 and 5?

Let T be the set of all triangles in a plane with R a relation in T given by R={(T_1,T_2): T_1~=T_2} . Show that R is an equivalence relation.

Let T be the set of all triangles in a plane with R a relation in T given by R={(T_1,T_2): T_1" is congruent to "T_2} . Show that R is an equivalence relation.

Let "T" be the set of all triangles in a plane with "R" as relation in "T" given by "R"={("T"_1," T"_2): T_1~="T"_2} . Show that "R" is an equivalence relation.

If R is a relation on the set T of all triangles drawn in a plane defined by a R b iff a is congruent to b for all, a, b in T, then R is

(2-4t+5t^2)-(3t^2 + 2t -7) is equivalent to :

Isosceles triangle T_1 has a base of 12 meters and a height of 20 meters . The vertices of a second triangle T_2 are the midpoints of the sides of T_1 . The vertices of a third triangle , T_3 , are the midpoints of the sides of T_2 . Assume the process continues indefinitely , with the vertices of T_(k+1) being the midpoints of the sides of T_k for every positive integer k. What is the sum of the areas, in square meters, of T_1,T_2,T_3 , ..... ?