Constructions, Congruence and Transformations

Constructions must be done with ruler and compass only — no protractor. Leave all arcs visible in the final answer.

Perpendicular bisector of a line segment $AB$:

1. Open the compass to more than half the length of $AB$.
2. Draw arcs above and below the line from $A$ and $B$ (same compass setting).
3. Connect the two intersection points — this is the perpendicular bisector.

Result: every point on the perpendicular bisector is equidistant from $A$ and $B$.

Bisecting an angle at $P$:

1. Draw an arc centred at $P$ that crosses both arms of the angle at $X$ and $Y$.
2. Draw equal arcs from $X$ and $Y$ that intersect at $Z$.
3. Draw $PZ$ — this is the angle bisector.

Perpendicular from a point to a line (and at a point on a line): use the same principle — place the compass at the point, find two equal-distance points on the line, then bisect the resulting segment.

Shortest distance from a point to a line = the perpendicular distance. Any other path to the line is longer.

A locus (plural: loci) is the set of all points satisfying a given condition.

Condition	Locus
Fixed distance from point $P$	Circle centred at $P$
Equal distance from two points $A$, $B$	Perpendicular bisector of $AB$
Fixed distance from a line	Two lines parallel to the original
Equal distance from two lines	Angle bisector of the angle between the lines

Worked example — shade the region less than 3 cm from point $P$ and closer to $A$ than to $B$.

Draw a circle of radius 3 cm centred at $P$; draw the perpendicular bisector of $AB$. Shade the intersection of the interior of the circle and the $A$-side of the bisector.

Two triangles are congruent (identical in shape and size) if any one of these conditions holds:

Criterion	What it means
SSS	All three sides equal
SAS	Two sides and the included angle equal
ASA	Two angles and the included side equal
RHS	Right angle, hypotenuse, and one other side equal

AAA is not a congruence criterion — it only guarantees similarity (same shape, possibly different size).

G6 — isosceles triangle: the base angles of an isosceles triangle are equal. Proof: draw the perpendicular bisector from the apex to the base; two congruent right-angled triangles (RHS) imply the base angles are equal.

Pythagoras as derivation (G6): from the areas of squares on each side of a right-angled triangle, $a^2+b^2=c^2$. Used to prove lengths and check right angles.

The four standard transformations are rotation, reflection, translation and enlargement.

Transformation	Fully described by	Preserves size?	Preserves shape?
Translation	Vector $\left(\genfrac{}{}{0ex}{}{a}{b}\right)$	Yes	Yes
Rotation	Centre, angle, direction	Yes	Yes
Reflection	Mirror line equation	Yes	Yes
Enlargement	Centre and scale factor	No	Yes

Enlargement with fractional scale factor ($0<k<1$): the image is smaller than the object; all lengths are multiplied by $k$.

Enlargement with negative scale factor (Higher): the image appears on the opposite side of the centre from the object and is also rotated by 180°. If scale factor $=-2$, each point is twice as far from the centre but on the opposite side.

Coordinate enlargement — enlarge point $(x,y)$ from centre $(cx,cy)$ by factor $k$: $\text{Image}=(cx+k(x-cx),cy+k(y-cy))$

A combined transformation applies two or more transformations in sequence. The order matters — the result may differ if the order is swapped.

Invariant points are points that map to themselves under a transformation.

• Under a reflection: all points on the mirror line are invariant.
• Under a rotation: only the centre of rotation is invariant (if it lies on the object).
• Under a translation: no points are invariant (unless the vector is zero).
• Under an enlargement with $k\ne 1$: only the centre of enlargement is invariant.

Worked example — reflect shape $S$ in the $y$-axis to get $S^{\prime }$, then rotate $S^{\prime }$ by $90°$ clockwise about the origin to get $S^{\prime \prime }$. Describe the single transformation that maps $S$ to $S^{\prime \prime }$.

After reflection in $y$-axis: $(x,y)\to (-x,y)$. After $90°$ clockwise rotation: $(-x,y)\to (y,x)$. Net effect: $(x,y)\to (y,x)$ — a reflection in the line $y=x$. ✓

(Extra context — any two reflections in perpendicular lines combine to give a rotation of 180° about their intersection; two reflections in parallel lines give a translation.)

1. Congruence — using AAA as a criterion

AAA guarantees similarity, not congruence. Two triangles can have all the same angles but be different sizes. A congruence proof must use SSS, SAS, ASA, or RHS.

2. Describing a rotation — omitting direction or centre

A rotation needs three things: the angle, the direction (clockwise/anticlockwise), and the centre. Missing any one of these loses marks.

3. Enlargement — measuring from the wrong point

Enlargement is measured from the centre of enlargement, not from the origin or from the shape. If the centre is at $(2,3)$ and scale factor 2, move each vertex twice as far from $(2,3)$, not twice as far from $(0,0)$.

4. Constructions — erasing arcs

Construction arcs must remain visible — erasing them removes the evidence that the construction was done correctly.

Mistake	Correction
"Enlarge by scale factor $-2$ from $(0,0)$: point $(3,1)$ → $(6,2)$"	With negative scale factor: $(3,1)\to (-6,-2)$ (multiply and negate)
"SAS: sides 4, 5 and angle of 60° between them ≡ sides 4, 5 and angle of 60° adjacent to the 4-side"	SAS requires the angle to be between (included by) the two given sides — different configurations are not necessarily congruent
"Perpendicular bisector: draw a line crossing through the midpoint of $AB$"	The perpendicular bisector must be perpendicular to $AB$ and pass through its midpoint — use compass arcs to guarantee this