NCERT Exemplar Solutions
Class 6 - Mathematics - Unit 9: Symmetry and Practical Geometry - Problems and Solutions

Question 70

Explanation (Very Beginner Friendly)

Given: In the figure, point C is the mirror image of A in line l. The segment BC cuts line l at P.

Key mirror fact (to remember): When you reflect a point across a line (like a mirror), any point on the mirror line is equally far from the original and its image.

(a) Is the image of P in line l the point P itself?

1. P lies on the mirror line l.
2. A point on the mirror line does not move under reflection.
3. So, the image of P is P itself.

(b) Is (PA = PC)?

1. C is the mirror image of A in line l.
2. P is on line l, i.e., on the mirror.
3. For any point on the mirror line, distances to a point and its image are equal.
4. Therefore, (PA = PC).

(c) Is (PA + PB = PC + PB)?

1. From part (b), we already have (PA = PC).
2. Add the same length (PB) to both sides.
3. Equality stays true when you add the same quantity to both sides.

(d) Is P the point on line l from which the sum of the distances of A and B is minimum?

1. We want to make the path A → (some point on l) → B as short as possible.
2. Use the reflection trick: replace A by its mirror image C across line l.
3. For any point (X in l), the “broken” path length is

   ( AX + XB = CX + XB )

   because (AX = CX) when (X) is on the mirror.
4. Shortest path between two points is a straight line. So among all choices of (X) on l, the smallest value of (CX + XB) happens when (X) lies on the straight line from C to B.
5. That straight line CB already meets the mirror at P (given).
6. Therefore, the minimum of (AX + XB) occurs at P.

Final Answers: (a) Yes   (b) Yes   (c) Yes   (d) Yes