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

Question 69

Solution (Very Beginner-Friendly)

We will first draw the mirror images of the points A and B in the line l. Then we will measure the lengths and compare.

1. Draw A′ (image of A in line l)
   1. With a set square or protractor, draw a short line from A to l that is perpendicular to l. Let it meet l at point P.

   ( AP perp l )

   2. Measure the distance AP with a ruler.
   3. On the other side of line l, along the same perpendicular, mark a point A′ so that the distance from P to A′ is the same as AP.

   ( PA' = PA )

2. Draw B′ (image of B in line l)
   1. Draw a perpendicular from B to l. Let it meet l at Q.

   ( BQ perp l )

   2. Measure the distance BQ.
   3. On the other side of l, along the same perpendicular, mark B′ so that the distance from Q to B′ equals BQ.

   ( QB' = BQ )

3. Measure the two segment lengths
   1. Use a ruler to measure the length AB.
   2. Use the ruler again to measure the length A′B′.
4. Compare

   ( AB stackrel{?}{=} A'B' )

   You will find that the two lengths are equal.

   ( AB = A'B' )

Why are they equal?

• Reflecting (taking a mirror image) in a line is a distance-preserving move. This is also called an isometry.
• Each point and its image are the same distance from the mirror line and lie on a line perpendicular to it.

( ext{dist}(A,l) = ext{dist}(A',l) )

( ext{dist}(B,l) = ext{dist}(B',l) )

• Because the whole figure is “flipped” without stretching, the distance between A and B is the same as the distance between A′ and B′.

( herefore; AB = A'B' )

Note: If your measured values differ slightly, it is usually due to drawing or measuring errors. Re-check the perpendiculars and equal distances from the line l.

Answer: Yes, ( AB = A'B' ).