Constructions & Loci flashcards
9 cards on Constructions & Loci: the key points, worked examples and the slips to watch. Swipe through them, or print them.
Card 1 of 9
Constructions & Loci
The perpendicular bisector
How do you construct the perpendicular bisector of a line AB with a ruler and compasses?
Answer it in your head, then turn the card over.
The perpendicular bisector
Key points
- Set the compasses to more than half the length of AB.
- Draw arcs from A and from B, above and below the line, without changing the setting.
- Join the two points where the arcs cross with a ruler.
- The line cuts AB in half at right angles, and every point on it is as far from A as from B.
Watch out
Rubbing out the arcs: they are your working, and without them even a perfect line can score nothing.
Constructions & Loci
The angle bisector
How do you construct the bisector of an angle with a ruler and compasses?
Answer it in your head, then turn the card over.
The angle bisector
Key points
- Put the compass point on the vertex and draw an arc that cuts both arms.
- From each of those two cuts, draw arcs of the same radius that cross inside the angle.
- Join the vertex to the crossing point: the line cuts the angle into two equal halves.
- Check it with a protractor if there is time, but leave every arc showing.
Watch out
Changing the compass setting between the two crossing arcs pulls the line off the middle.
Constructions & Loci
A triangle from three sides
How do you construct a triangle when you know the lengths of all three sides?
Answer it in your head, then turn the card over.
A triangle from three sides
Key points
- Draw one side as the base with a ruler.
- Set the compasses to the second length and draw an arc from one end of the base.
- Set them to the third length and draw an arc from the other end.
- Where the arcs cross is the third corner: join it to both ends of the base.
Sides 7 cm, 5 cm and 4 cm
Base 7 cm; an arc of radius 5 cm from one end, 4 cm from the other
The arcs cross because 5 + 4 = 9 cm is more than 7 cm
Watch out
Rubbing out the arcs: they are what shows the triangle was constructed, not drawn with a protractor.
Constructions & Loci
Constructing 60° and 90°
How do you construct angles of 60° and 90° with only a ruler and compasses?
Answer it in your head, then turn the card over.
Constructing 60° and 90°
Key points
- 60°: draw an arc from the end of a line, then the same radius from where it cuts the line; join the end to the crossing.
- It works because the three points make an equilateral triangle.
- 90° at a point on a line: mark two points the same distance either side of it, then construct their perpendicular bisector.
Why, and the angles that follow
Each angle of an equilateral triangle: 180° ÷ 3 = 60°
Bisect 60° for 30°: 60° ÷ 2 = 30°
Bisect 90° for 45°: 90° ÷ 2 = 45°
Watch out
Changing the radius between the two arcs of a 60° construction: the triangle is no longer equilateral.
Constructions & Loci
Loci a fixed distance away
What does the locus of points a fixed distance from a point, or from a line segment, look like?
Answer it in your head, then turn the card over.
Loci a fixed distance away
Key points
- A locus is the set of all the points that follow a rule.
- A fixed distance from a point: a circle, with the point as its centre.
- A fixed distance from a line segment: a running track, two parallel lines joined by semicircles centred on its ends.
Watch out
Two parallel lines on their own leave out the points round the ends: draw the semicircles too.
Constructions & Loci
Loci the same distance from two things
What is the locus of points the same distance from two points, and from two lines?
Answer it in your head, then turn the card over.
Loci the same distance from two things
Key points
- The same distance from two points A and B: the perpendicular bisector of AB.
- The same distance from two lines that meet: the bisector of the angle between them.
- Two points, perpendicular bisector; two lines, angle bisector.
Watch out
Using an angle bisector for two points is the most common slip in loci questions.
Constructions & Loci
Shading regions
How do you shade the region of points that meets several conditions at once?
Answer it in your head, then turn the card over.
Shading regions
Key points
- Draw each condition's boundary: a circle for "within" a distance, a bisector for "closer to A than B".
- Decide the side for each: within means inside the circle; closer to A means A's side of the bisector.
- Shade only where every condition holds, then test a point in your region against each one.
Within 4 m of tap T, closer to A than B
Scale 1 cm to 2 m: a circle round T of radius 4 ÷ 2 = 2 cm
Construct the perpendicular bisector of AB
Shade inside the circle, on A's side of the bisector
Watch out
Shading the side that breaks a condition: further from A is the other side of the bisector.
Constructions & Loci
Using a scale
How do you use a scale such as 1 cm to 4 m on a constructions or loci drawing?
Answer it in your head, then turn the card over.
Using a scale
Key points
- Real life to the drawing: divide by the scale before you set the compasses.
- The drawing to real life: multiply by the scale.
- Measure to the nearest millimetre: a small slip on paper is a big one in real life.
Scale 1 cm to 4 m
A 14 m rope: 14 ÷ 4 = 3.5 cm on the compasses
Two points 6.2 cm apart: 6.2 × 4 = 24.8 m in real life
Watch out
Setting the compasses to the real distance: at 1 cm to 4 m, 14 m is 3.5 cm, not 14 cm.
Constructions & Loci
Scales written as a ratio
How do you use a scale written as a ratio, such as 1 : 500?
Answer it in your head, then turn the card over.
Scales written as a ratio
Key points
- 1 : 500 means every real length is 500 times the length on the drawing.
- Both sides are in the same unit: 1 cm on the drawing is 500 cm in real life, which is 5 m.
- Change cm to m at the end by dividing by 100.
Scale 1 : 500
3.6 cm on the drawing: 3.6 × 500 = 1800 cm, which is 1800 ÷ 100 = 18 m
A real 12 m is 1200 cm: 1200 ÷ 500 = 2.4 cm on the drawing
Watch out
Mixing units: 3.6 × 500 gives centimetres, not metres, so 1800 is 18 m.
Print these cards
Four cards to an A4 sheet, each the size of a standard A6 revision card.
One-sided
Each card prints on one face: the question at the top, the notes under it.
Two-sided
Questions on the front, notes on the back. Print double-sided and flip on the long edge, then cut along the lines.
Put it into practice
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