What if, x1, x2, y1, y2 take negative values? In the figure, triangle AMD is reflected in the y-axis. What are the coordinates of the images of points A, M, and D?
Step-by-step solution
Idea: A point and its mirror image are the same distance from the mirror, on opposite sides, on a line at right angles to the mirror. For the y-axis as mirror this means (x, y) → (−x, y).
- Reflecting in the y-axis: the image is as far to the left of the y-axis as the point is to the right, at the same height. So the y-coordinate stays the same and the x-coordinate changes sign: (x, y) → (−x, y).½ mark
- A (3, 4) → A′ (−3, 4); M (9, 6) → M′ (−9, 6); D (7, 1) → D′ (−7, 1).1½ marks
- Now the coordinates are negative. Use C′ (−3, 1): C′D′ = −3 − (−7) = 4 and A′C′ = 4 − 1 = 3, so A′D′ = √(42 + 32) = 5 units.½ mark
- Similarly D′M′ = √((−9 − (−7))2 + (6 − 1)2) = √((−2)2 + 52) = √29 units and M′A′ = √((−3 − (−9))2 + (4 − 6)2) = √(62 + (−2)2) = √40 units. The negative values cause no trouble, because each shift is squared.½ mark
Check: Each point and its image are equally far from the y-axis: A and A′ are both 3 units away, M and M′ 9, D and D′ 7 ✓.
Answer to write in the exam
Reflection in the y-axis: (x, y) → (−x, y)
A (3, 4) → A′ (−3, 4)
M (9, 6) → M′ (−9, 6)
D (7, 1) → D′ (−7, 1)
A′D′ = √((−3 + 7)2 + (4 − 1)2) = √25 = 5 units
D′M′ = √((−2)2 + 52) = √29 units; M′A′ = √(62 + (−2)2) = √40 units
∴ A′ (−3, 4), M′ (−9, 6), D′ (−7, 1); side lengths unchanged
Common mistakes that cost marks
- Changing the sign of the y-coordinate, e.g. A′ (3, −4). That is reflection in the x-axis, not the y-axis.
- Changing both signs, e.g. A′ (−3, −4). That point is the image after turning half a turn about O.
- In the distance working, writing −3 − (−7) = −10 instead of −3 + 7 = 4.
How this can come in the exam
The image of the point (−4, 7) on reflection in the y-axis is
- (4, 7)
- (−4, −7)
- (4, −7)
- (7, −4)
Show answer
(A) (4, 7)
Reflection in the y-axis changes the sign of the x-coordinate only.
Triangle PQR with P (1, 2), Q (5, 2) and R (5, 7) is reflected in the y-axis. Write the coordinates of the images and find P′R′.
Show answer
P′ (−1, 2), Q′ (−5, 2), R′ (−5, 7) (1 mark). P′R′ = √((−5 + 1)2 + (7 − 2)2) = √(16 + 25) = √41 units, equal to PR (1 mark).Try one yourself
Reflect A (2, 3), B (6, 3) and C (2, 9) in the y-axis. Write the images and find B′C′.
Show answer
A′ (−2, 3), B′ (−6, 3), C′ (−2, 9). B′C′ = √(42 + 62) = √52 = 2√13 units.
More questions like this
- 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)? - What are the x-coordinate and y-coordinate of the point of intersection of the two axes?
- Point W has x-coordinate equal to −5. Can you predict the coordinates of point H which is on the line through W parallel to the y-axis? Which quadrants can H lie in?
- Consider the points R (3, 0), A (0, −2), M (−5, −2) and P (−5, 2). If they are joined in the same order, predict:
- Plot point Z (5, −6) on the Cartesian plane. Construct a right-angled triangle IZN and find the lengths of the three sides.
(Comment: Answers may differ from person to person.)