Math 7 · Constructions and transformations
Lines you can trust, and shapes on the move
A construction draws a line that is guaranteed to be exactly right — perpendicular, parallel, or cutting something exactly in half — with a ruler, a compass or a fold. Then the Cartesian plane gives every point an address, so you can say exactly how a shape slides, flips or turns.
- 1. The words, first
- 2. Parallel and perpendicular lines and bisectors
- 3. The Cartesian plane
- 4. Translations, reflections and rotations
- 5. What costs marks
The words, first
The idea: Two groups of words: the lines you construct, and the moves you make on the grid.
| Word | What it means |
|---|---|
| Line segment | The part of a line between two endpoints, such as AB. |
| Parallel | Lines on a flat surface that never meet. They stay the same distance apart. |
| Perpendicular | Meeting at 90°, a square corner. |
| Midpoint | The point exactly halfway along a segment. |
| Vertex, arms | The corner point of an angle, and the two lines that meet there. |
| Bisector | A line that cuts something into two equal parts. A perpendicular bisector cuts a segment in half at 90°. An angle bisector cuts an angle into two equal angles. |
| Cartesian plane | A grid made by a horizontal x-axis and a vertical y-axis crossing at the origin, (0, 0). The axes make four quadrants, numbered I to IV counterclockwise — the opposite way to a clock — from the top right. |
| Ordered pair (x, y) | A point's address: x across from the origin (right if positive, left if negative), then y up or down. The two numbers are its coordinates. |
| Image | A shape after a move. Its points get a prime mark: A becomes A′, read “A prime”. |
| Translation, reflection, rotation | A slide; a flip over a mirror line; a turn about a fixed point. |
Parallel and perpendicular lines and bisectors
The idea: Each construction comes with a guarantee. A perpendicular bisector is every point the same distance from both ends of a segment. An angle bisector splits an angle exactly in half.
A perpendicular bisector of AB, with a compass.
- Open the compass to more than half the length of AB.
- With the point on A, draw an arc above the segment and another below it.
- Without changing the compass, do the same from B.
- Join the two points where the arcs cross.
What it guarantees. The line crosses AB at its midpoint, at 90°, and every point on it is the same distance from A as from B — which is why equal arcs from A and B cross on it. Folding the paper so A lands on B gives the same line as a crease.
Worked example. Two trees are 8 m apart. Where can a bench go so it is the same distance from both? Anywhere on the perpendicular bisector of the line between them — closest at the midpoint, 4 m from each.
Angle bisector. With the compass point on the vertex, draw an arc crossing both arms. From each crossing, draw equal arcs that meet inside the angle. A line from the vertex through that meeting point splits the angle in two: 70° becomes two angles of 35°.
Parallel lines. Two lines both perpendicular to the same line are parallel, so construct two perpendiculars to one line.
The Cartesian plane
The idea: An ordered pair is an address: x first, across; then y, up or down. The signs tell you the quadrant before you plot anything.
| Quadrant | x is | y is | Example |
|---|---|---|---|
| I | positive | positive | (3, 4) |
| II | negative | positive | (−3, 2) |
| III | negative | negative | (−5, −3) |
| IV | positive | negative | (2, −1) |
Worked example: plot (−3, 2). From the origin, the first number is −3, so go 3 left. The second is 2, so go 2 up. The point is in quadrant II. Going 3 up and 2 left lands on (−2, 3) instead: the order matters.
On an axis, in no quadrant. (0, −4) is on the y-axis, so it is in no quadrant.
Translations, reflections and rotations
The idea: A translation slides every point the same way. A reflection flips the shape over a line. A rotation turns it about a point. Compare the coordinates before and after to tell which happened.
| Start | Slide | Flip | Turn |
|---|---|---|---|
| A (2, 1) | (−5, −3) | (2, −1) | (−1, 2) |
| B (5, 1) | (−2, −3) | (5, −1) | (−1, 5) |
| C (2, 3) | (−5, −1) | (2, −3) | (−3, 2) |
- Translation, 7 left and 4 down. Every x went down by 7 and every y by 4: A (2, 1) became (2 + (−7), 1 + (−4)) = (−5, −3). Same size, same shape, facing the same way.
- Reflection in the x-axis. Each x stays and each y becomes its opposite, so every point lands the same distance away on the other side of the axis. The image is mirror-reversed: A to B to C now runs clockwise instead of counterclockwise.
- Rotation, 90° counterclockwise about the origin. The shape is turned, not flipped, so A to B to C still runs counterclockwise. Each point swings round the origin and stays the same distance from it. For this quarter turn, each image point is the original swapped, with the new first coordinate made opposite: B (5, 1) becomes (−1, 5).
Half a turn. A 180° rotation about the origin makes both coordinates opposite: (2, 1) becomes (−2, −1).
Naming a move from before and after. Flipped? A reflection. If not, did every point move by the same amounts? A translation. Otherwise, a rotation.
What costs marks
The idea: Most of these are reading the grid the wrong way round.
- Swapping x and y. (−3, 2) is 3 left, then 2 up.
- Changing the wrong coordinate in a reflection. Reflecting in the x-axis changes y, because the point crosses that axis going up or down.
- Turning the wrong way. Counterclockwise is the opposite way to a clock.
- Giving a point on an axis a quadrant.
- Opening the compass to less than half the segment. The arcs never meet.