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:
- (i) Two sides of RAMP that are perpendicular to each other.
- (ii) One side of RAMP that is parallel to one of the axes.
- (iii) Two points that are mirror images of each other in one axis. Which axis will this be?
- Verify Now plot the points and verify your predictions.
Step-by-step solution
Idea: Two points with the same y-coordinate lie on a horizontal line (parallel to the x-axis); two with the same x-coordinate lie on a vertical line (parallel to the y-axis). Horizontal and vertical lines are perpendicular. Points (x, y) and (x, −y) are mirror images in the x-axis.
(i) Two sides of RAMP that are perpendicular to each other.
- A (0, −2) and M (−5, −2) have the same y-coordinate, so AM is horizontal. M (−5, −2) and P (−5, 2) have the same x-coordinate, so MP is vertical.½ mark
- A horizontal line and a vertical line meet at a right angle, so AM ⊥ MP (the right angle is at M).½ mark
(ii) One side of RAMP that is parallel to one of the axes.
- AM is horizontal (y = −2 throughout), so AM is parallel to the x-axis. (Equally, MP is parallel to the y-axis.)½ mark
(iii) Two points that are mirror images of each other in one axis. Which axis will this be?
- M (−5, −2) and P (−5, 2) have the same x-coordinate and y-coordinates that are equal but opposite in sign. So they are the same distance (2 units) below and above the x-axis, on the same vertical line.½ mark
- So M and P are mirror images of each other in the x-axis. (No other pair works: R and A have no matching partner.)½ mark
Verify Now plot the points and verify your predictions.
- On the plot (see the figure), AM runs along the line y = −2 and MP runs straight up along x = −5, making a right angle at M; P sits directly above M, 2 units on the other side of the x-axis. RA and PR are slanted, so they are not parallel to either axis. All three predictions are confirmed.½ mark
Check: Right angle at M by the Baudhāyana–Pythagoras theorem: AM = 5, MP = 4, AP = √(52 + 42) = √41, and AM2 + MP2 = 25 + 16 = 41 = AP2 ✓.
Answer to write in the exam
(i)
A (0, −2), M (−5, −2): same y-coordinate ⇒ AM horizontal
M (−5, −2), P (−5, 2): same x-coordinate ⇒ MP vertical
∴ AM ⊥ MP
(ii)
AM: y = −2 at both ends
∴ AM ∥ x-axis (also MP ∥ y-axis)
(iii)
M (−5, −2) and P (−5, 2): same x, y-coordinates opposite
∴ M and P are mirror images in the x-axis
Verify
Plotted: AM along y = −2, MP along x = −5, right angle at M
P is 2 units above the x-axis, M is 2 units below it, on the same vertical line
∴ Predictions (i), (ii), (iii) verified.
Common mistakes that cost marks
- Joining the points in a different order (e.g. R → M), which gives different sides. Join R → A → M → P → R.
- Saying M and P are mirror images in the y-axis. They are one above the other, so the mirror is the x-axis.
- Calling RA perpendicular to AM because A is on an axis. RA is slanted; only AM and MP are along grid lines.
How this can come in the exam
Which pair of points are mirror images of each other in the y-axis?
- (2, 5) and (−2, 5)
- (2, 5) and (2, −5)
- (2, 5) and (5, 2)
- (2, 5) and (−2, −5)
Show answer
(A) (2, 5) and (−2, 5)
Mirror images in the y-axis have the same y-coordinate and opposite x-coordinates.
The points K (−4, 1), L (2, 1) and N (2, −3) are joined in order. Which side is parallel to the x-axis, and which two sides are perpendicular?
Show answer
K and L have y = 1, so KL ∥ x-axis (1 mark). L and N have x = 2, so LN is vertical; hence KL ⊥ LN (1 mark).Try one yourself
For E (−1, 3), F (4, 3) and G (4, −3) joined in order, name two perpendicular sides and two points that are mirror images in an axis.
Show answer
EF ⊥ FG (EF horizontal, FG vertical). F (4, 3) and G (4, −3) are mirror images in the x-axis.
More questions like this
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