Question

1. Point P(2, –3) on reflection becomes P’(2, 3). Name the line of reflection (say L₁).
2. Point P’ is reflected to P” along the line (L₂), which is perpendicular to the line L₁ and passes through the point, which is invariant along both axes. Write the coordinates of Р”.
3. Name and write the coordinates of the point of intersection of the lines L₁ and L₂.
4. Point P is reflected to P” on reflection through the point named in the answer of part I of this question. Write the coordinates of P”. Comment on the location of the points P” and P”.

Answer 1

Given: P = (2, –3) and its image after one reflection P’ = (2, 3).

Step-wise calculation: 

a. Find the line of reflection L₁.

Observation: x-coordinate is unchanged (2 → 2) while y changes sign (–3 → 3). 

That is the rule for reflection in the x‑axis: Mx(x, y) = (x, –y). 

Therefore, L₁ is the x‑axis, i.e. y = 0.

b. Find L₂ perpendicular to L₁ and passing through the point invariant under both axes and the coordinates of P” image of P’ in L₂.

The point invariant under both the x‑ and y‑axis reflections is the origin O = (0, 0).

A line perpendicular to L₁ the x‑axis and through O is the y‑axis, x = 0.

So, L₂ is the y‑axis.

Reflection in the y‑axis maps (x, y) → (–x, y).

Apply this to P’ = (2, 3): P” = (–2, 3).

c. Intersection of L₁ and L₂.

L₁: y = 0 and L₂: x = 0 intersect at the origin O = (0, 0)

d. Reflect P through the point named in (c) the origin to get P’” and comment on P” and P’”.

Reflection through the origin maps (x, y) → (–x, –y).

Apply to P = (2, –3): P’” = (–2, 3).

Compare: P” = (–2, 3) and P’” = (–2, 3) they are the same point.

Comment: Both P” and P’” lie at (–2, 3), which is in quadrant II; P” was obtained by successive reflections in the x‑axis then the y‑axis, while P’” is the single point reflection of P through the origin these give the same result.