Question

In the below figure, point C is the image of point A in line l and line segment BC intersects the line l at P.

(a) Is the image of P in line l the point P itself?
(b) Is PA = PC?
(c) Is PA + PB = PC + PB?
(d) Is P that point on line l from which the sum of the distances of points A and B is minimum?

Answer 1

Find the image of point P in line l in the given figure.
Yes, the image of P in line l is the point P itself, as it lies on the line of symmetry.
We know that by symmetry a point and its image are at equal distance from the line of symmetry.

Hence, the image of P in line l is the point P itself.

(b) Check if PA = PC.
By symmetry a point and its image are at equal distance from the line of symmetry.
Also, any point on this line of symmetry will be at equal distance from the point and its image.
Thus, PA = PC. [ Since P is a point on the line of symmetry]

Hence, PA = PC.

(c) Check if PA + PB = PC + PB.
We know that PA = PC [ Since P is a point on the line of symmetry]
Now, add PB on both the sides of the above expression.
As adding same quantity would not change the equality of an equation.
So, PA + PB = PC + PB

Hence, PA + PB = PC + PB.

(d) Check the given statement and find the shortest distance between A & B.
Given,
Is P that point on line l from which the sum of the distances of points A and B is minimum?
Yes, since PA + PB = CP + PB [C is image of P].

Now,
CB is the shortest path between C and B
But CB = CP + PB
[ Since P is a point on line CB]
= AP + PB [ Since AP = CP]
Hence, P is the point such that AP + PB is the shortest distance or minimum.