Let R be the relation "is congruent to" on the set of all triangles in a plane. Is R:
Show Hint
- Reflexive only
- Symmetric only
- Symmetric and reflexive only
- Equivalence relation
The Correct Option is D
Solution and Explanation
Step 1: {Check for Reflexivity}
A relation is reflexive if every element is related to itself. In this case, every triangle is congruent to itself. So, \( \triangle A \cong \triangle A \). Thus, the relation R is reflexive.
Step 2: {Check for Symmetry}
A relation is symmetric if for every \( a \) related to \( b \), \( b \) is also related to \( a \). If \( \triangle A \cong \triangle B \), then \( \triangle B \cong \triangle A \). Thus, the relation R is symmetric.
Step 3: {Check for Transitivity}
A relation is transitive if whenever \( a \) is related to \( b \) and \( b \) is related to \( c \), then \( a \) is also related to \( c \). If \( \triangle A \cong \triangle B \) and \( \triangle B \cong \triangle C \), then \( \triangle A \cong \triangle C \). Thus, the relation R is transitive.
Step 4: {Conclusion}
Since the relation R is reflexive, symmetric, and transitive, it is an equivalence relation. Therefore, the correct answer is (D).
Top Questions on Plane
- Let the foot of perpendicular from a point \( P(1,2,-1) \) to the straight line \( L : \frac{x}{1} = \frac{y}{0} = \frac{z}{-1} \) be \( N \). Let a line be drawn from \( P \) parallel to the plane \( x + y + 2z = 0 \) which meets \( L \) at point \( Q \). If \( \alpha \) is the acute angle between the lines \( PN \) and \( PQ \), then \( \cos \alpha \) is equal to:
- Let the acute angle bisector of the two planes \( x - 2y - 2z + 1 = 0 \) and \( 2x - 3y - 6z + 1 = 0 \) be the plane \( P \). Then which of the following points lies on \( P \)?
- If one of the lines given by \( 6x^2 - xy + 4cy^2 = 0 \) is \( 3x + 4y = 0 \), then \( c \) equals
- The equation of the plane passing through the point \( (1, 1, 1) \) and perpendicular to the planes \( 2x + y - 2z = 5 \) and \( 3x - 6y - 2z = 7 \) is:
- The distance between the two planes $2x + 3y + 4z = 4$ and $4x + 6y + 8z = 12$ is:
Questions Asked in BITSAT exam
- For two events A and B, if \(P(A) = P(A/B) = \frac{1}{4}\) and \(P(B/A) = \frac{1}{2}\), then which of the following is not true?
- Given below is the distribution of a random variable \(X\):
\[ \begin{array}{|c|c|} \hline X = x & P(X = x) \\ \hline 1 & \lambda \\ 2 & 2\lambda \\ 3 & 3\lambda \\ \hline \end{array} \] If \(\alpha = P(X<3)\) and \(\beta = P(X>2)\), then \(\alpha : \beta = \)- BITSAT - 2024
- Probability
- The probability of getting 10 in a single throw of three fair dice is:
- BITSAT - 2024
- Probability
- If \( \vec{a} = 2\hat{i} + \hat{j} + 2\hat{k} \), then the value of \( |\hat{i} \times (\vec{a} \times \hat{i})| + |\hat{j} \times (\vec{a} \times \hat{j})| + |\hat{k} \times (\vec{a} \times \hat{k})|^2 \) is equal to:}
- BITSAT - 2024
- Algebra
- Let \( \mathbf{a} = \hat{i} - \hat{k}, \mathbf{b} = x\hat{i} + \hat{j} + (1 - x)\hat{k}, \mathbf{c} = y\hat{i} + x\hat{j} + (1 + x - y)\hat{k} \). Then, \( [\mathbf{a} \, \mathbf{b} \, \mathbf{c}] \) depends on:}
- BITSAT - 2024
- Vectors