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Reflection in the axes · 3 marks

1. What has remained the same and what has changed with this reflection?
2. Would these observations be the same if ΔADM is reflected in the x-axis (instead of the y-axis)?

  1. 1. What has remained the same and what has changed with this reflection?
  2. 2. Would these observations be the same if ΔADM is reflected in the x-axis (instead of the y-axis)?
Answer: 1. Same: side lengths (5, √29, √40), angles, shape, size and area (13 sq units), and every y-coordinate. Changed: every x-coordinate changes sign, the triangle moves from Quadrant I to Quadrant II, and it is flipped (mirror image). 2. Yes, the same kind of observations: lengths, angles and area stay the same, but now the x-coordinates stay, the y-coordinates change sign, and the triangle moves to Quadrant IV.

Step-by-step solution

Given: Triangle ADM: A (3, 4), D (7, 1), M (9, 6); Its reflection in the y-axis: A′ (−3, 4), D′ (−7, 1), M′ (−9, 6)

Idea: A reflection moves a figure without stretching or squashing it, so all lengths and angles are kept. Only the position changes: the coordinate measured across the mirror changes sign.

xy−9−6−3369−6−336A (3, 4)D (7, 1)M (9, 6)A″ (3, −4)D″ (7, −1)M″ (9, −6)image inthe y-axis

1. What has remained the same and what has changed with this reflection?

  1. Remained the same: the side lengths (AD = A′D′ = 5, DM = D′M′ = √29, MA = M′A′ = √40), so the angles, shape, size and area (13 square units) are the same; the triangles are congruent. Each y-coordinate and each vertex’s distance from the y-axis are also unchanged.1 mark
  2. Changed: each x-coordinate has changed sign (3 → −3, 7 → −7, 9 → −9); the triangle has moved from Quadrant I to Quadrant II; and it is flipped: going A → D → M is anticlockwise, but A′ → D′ → M′ is clockwise.½ mark
Lengths, angles, shape, area and the y-coordinates stay the same; the x-coordinates change sign, the quadrant changes from I to II and the triangle is flipped.

2. Would these observations be the same if ΔADM is reflected in the x-axis (instead of the y-axis)?

  1. Reflection in the x-axis: (x, y) → (x, −y). A″ (3, −4), D″ (7, −1), M″ (9, −6).½ mark
  2. A″D″ = √(42 + 32) = 5, D″M″ = √(22 + 52) = √29, M″A″ = √(62 + 22) = √40. So again lengths, angles, shape and area are unchanged.½ mark
  3. Yes, the same kind of observations hold, with the roles of the axes swapped: now the x-coordinates stay the same and the y-coordinates change sign; the triangle moves to Quadrant IV, and it is again flipped.½ mark
Yes: lengths, angles and area are again unchanged; this time the x-coordinates stay, the y-coordinates change sign and the triangle moves to Quadrant IV.
1. Side lengths, angles, shape, size, area and the y-coordinates stay the same; the x-coordinates change sign, the triangle moves from Quadrant I to Quadrant II and is flipped. 2. Yes: reflecting in the x-axis also keeps lengths, angles and area, but now the y-coordinates change sign, the x-coordinates stay, and the triangle moves to Quadrant IV.

Check: Area of ADM by the box method: the 6 × 5 rectangle from (3, 1) to (9, 6) has area 30; remove the three corner triangles 6, 5 and 6: 30 − 17 = 13 sq units. The image triangles fit the same-sized box, so their area is also 13 ✓.

Answer to write in the exam

1.

Same: A′D′ = AD = 5, D′M′ = DM = √29, M′A′ = MA = √40 ⇒ angles, shape, area (13 sq units) same (triangles congruent)

Same: y-coordinates (4, 1, 6)

Changed: x-coordinates change sign; Quadrant I → Quadrant II; orientation reversed (mirror image)

2.

Reflection in the x-axis: (x, y) → (x, −y)

A″ (3, −4), D″ (7, −1), M″ (9, −6)

A″D″ = 5, D″M″ = √29, M″A″ = √40 ⇒ lengths, angles, area unchanged

∴ Yes; now x-coordinates unchanged, y-coordinates change sign, triangle in Quadrant IV.

Common mistakes that cost marks

  • Saying the lengths change because the coordinates became negative. Distances use squares of the shifts, so signs do not matter.
  • For the x-axis, changing the x-coordinates as before. The coordinate that changes sign is the one measured across the mirror, here y.
  • Saying “nothing has changed”. The position, the quadrant and the orientation (clockwise or anticlockwise order of the vertices) do change.

How this can come in the exam

MCQ (1 mark)

The image of the point (6, −2) on reflection in the x-axis is

  1. (6, 2)
  2. (−6, −2)
  3. (−6, 2)
  4. (−2, 6)
Show answer

(A) (6, 2)
Reflection in the x-axis keeps x and changes the sign of y.

Assertion–Reason (1 mark)

Assertion (A): The segment joining (1, 2) and (4, 6) and its reflection in the x-axis both have length 5 units.
Reason (R): Reflection in an axis does not change lengths.

  1. Both A and R are true, and R is the correct explanation of A.
  2. Both A and R are true, but R is not the correct explanation of A.
  3. A is true but R is false.
  4. A is false but R is true.
Show answer

(A) Both A and R are true, and R is the correct explanation of A.
Length = √(32 + 42) = 5; the image joins (1, −2) and (4, −6), also 5. R explains why.

Try one yourself

Reflect triangle P (−2, 1), Q (−5, 1), R (−2, 5) in the x-axis. Write the images, name the quadrant of the image and find Q′R′.

Show answer

P′ (−2, −1), Q′ (−5, −1), R′ (−2, −5), in Quadrant III. Q′R′ = √(32 + 42) = 5 units (= QR).

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