GCSE · Maths · Edexcel · Spec 1MA1 · Foundation+Higher

Standard ruler-and-compass constructions and loci

Two buoys, A and B. A boat is no closer to one than the other. Where could it be? Guess, then drag the boats.

Maths · Loci

Where could the boat be?
26.212.512.628.6134°ABMPQ

P to A 26.2. P to B 12.5. Q to A 12.6. Q to B 28.6. Angle AMP 134°. Relationship: Whenever a boat's two readings match, it is on the perpendicular bisector of AB: the straight line through M, at right angles to AB.

P to A26.2P to B12.5Q to A12.6Q to B28.6Angle AMP134°Slide P and Q until each boat's two readings match.

Whenever a boat's two readings match, it is on the perpendicular bisector of AB: the straight line through M, at right angles to AB.

Drag boats P and Q sideways. The chips show lengths on the diagram, rounded to one decimal place.

Maths · Angle bisector

The fence line between two walls

Put the angle-bisector construction in the order you would actually do it

1 · Do this first3 · Do this last

  1. Join the vertex to the point where those two arcs cross.

  2. Put the compass point on the vertex and draw an arc that crosses both arms.

  3. From each crossing point, draw arcs of the same radius so they cross inside the angle.

Watch out: Leave every arc on the page. The arcs are your working, and they are how someone can see the line is a construction.

Maths · Constructions

A perpendicular from a point on the line

Construct the perpendicular to a line at a point X that sits on the line.

  1. Put the compass point on X. Keeping one radius, draw an arc either side of X so it cuts the line at two marks. Call them S and T.X is now exactly halfway between S and T. Don't rub anything out.
  2. missing step
Which line is step 2?

Maths · Regions

Which boats are in the region?

Lighthouses A and B are 10 km apart. The region we want is closer to A than to B and within 6 km of A. Sort each boat using its distances.

Still to sort

Meets every condition (0)

On A's side of the perpendicular bisector and inside the 6 km circle.

On the bisector (0)

Exactly as far from A as from B.

Where the line is: A point exactly on the bisector is equidistant, so the bisector itself is the boundary.

Misses a condition (0)

Fails at least one of the two conditions.

Where the line is: Inside the circle is not enough on its own. A boat has to be on the right side of the bisector too.

6 of 6 still to sort.

Worked example · bearings and distance (invented numbers)

Problem

A boat leaves a port P on a bearing of 130° and sails at 40 km/h for 1.5 hours. On a scale drawing, 1 cm stands for 10 km. Where is the boat?

Maths · Loci with area

The mower and the corner

A mower is tied by a 5 m cord to a point on one wall of a building, 3 m from a corner. The grass carries on round that corner, the wall round the corner is longer than 2 m, and the other end of the first wall is more than 5 m away. Find the area of grass the mower can reach, leaving your answer in terms of π.

A student's answer — which line goes wrong?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Find that line with a ruler and compasses, then use it to find positions and regions.

What you need to know

  • A locus is the set of all points that satisfy a given condition. You find one by construction, leaving your construction marks visible.
  • A perpendicular bisector cuts a segment in half at right angles. It is the locus of points equidistant from the two end points.
  • Don't expect one point. Many points are equidistant from two points, and together they form the perpendicular bisector.
  • Have a goSam says: 'Only one place is exactly as far from buoy A as from buoy B: the middle.' Is Sam right?

    No. The midpoint is one of them, but so is every point on the perpendicular bisector.

    Equidistant points aren't a single point. Together they form a whole line, the perpendicular bisector.

  • To construct it, set the compasses to more than half the length. Draw same-radius arcs from each end, then join the crossings.
  • Have a goAB is 8 cm long and you set the compasses to 3 cm for its perpendicular bisector. What's wrong?

    3 cm is less than half of 8 cm. You need more than 4 cm.

    This construction needs the compasses set to more than half the length of the segment.

  • An angle bisector splits an angle into two equal parts. It is the locus of points equidistant from the two arms.
  • For a perpendicular at a point on a line, cut the line either side with equal arcs, then bisect the piece between.
  • Regions: build the boundary, then pick a side. Points exactly on a bisector are equidistant, so the bisector is the boundary.
  • A fixed distance from a point makes a circle: inside is closer, outside is further. Set the compasses to that distance, using the scale.
  • Have a goThe locus 'exactly 4 km from a lighthouse' is a circle. A boat is 3 km from the lighthouse. Inside the circle, on it, or outside?

    Inside. 3 km is closer than 4 km.

    Points inside the circle are closer than the circle's distance. Points outside it are further away.

  • A bearing is clockwise from North in three figures. It gives a ray, not a position: add distance = speed × time for the radius.
  • Around a barrier, like a building's corner, the locus is sectors of different radii. The area is the sum of the sectors.

The big picture

A locus is every point that fits a rule. Points equidistant from two points make the perpendicular bisector, points equidistant from two arms make the angle bisector, and with compasses, bearings and circles you can pin down the region that fits several rules at once.

Key points

1A locus is the set of all points that satisfy a given condition. You find loci with a ruler and a pair of compasses and leave the construction marks visible.
2The perpendicular bisector of a segment divides it into two equal parts at right angles. It is the locus of points equidistant from the two end points, and those points are many, not one.
3To construct a perpendicular bisector: compasses set to more than half the length, arcs of the same radius from each end crossing above and below, then join the crossings with a ruler. Where space is tight, draw two isosceles triangles on the same base and join their apex points.
4The angle bisector divides an angle into two equal parts and is the locus of points equidistant from the two arms. Build it with an arc across both arms, equal arcs from the two crossing points, then join the vertex to where those arcs cross.
5A perpendicular from a point on a line: equal arcs either side cut the line, then you construct the perpendicular bisector of the part between the cut marks.
6Regions: construct the boundary first and then choose a side. With several conditions, construct every boundary and shade only the region that satisfies all of them.
7A fixed distance from a point is a circle (inside is closer, outside is further). A bearing gives a ray, and where the circle meets the ray fixes the position.
8Around a barrier the locus is made of sectors with different radii, and the area is their sum, which can be left in terms of π.

Worked example

Problem

Construct the perpendicular bisector of a line segment AB that is 8 cm long, then say how you could check that a point on your line is the same distance from A and from B.

⚠ Watch out

Thinking there is only one point the same distance from two places (the midpoint). There are lots, and together they make the whole perpendicular bisector. A second slip: rubbing out the arcs. They are your working, so leave them in.

🧠

Memory hook

Equidistant from two points: perpendicular bisector. Equidistant from two arms: angle bisector. Bisect means 'cut in half', and a locus is just 'all the places that fit the rule'.

✓

Check yourself

A chest is buried as far from palm tree A as from palm tree B, and within 20 paces of A. What would you construct, and where could the chest be?

Flashcards

(16)
What is a locus?
The set of all points that satisfy a given condition.
What do you leave visible when you construct a locus?
The construction marks (your arcs).
What does a perpendicular bisector do to a line segment?
It divides the segment into two equal parts at right angles.
Points equidistant from two points lie on which line?
The perpendicular bisector of the segment joining the two points.
Is there only one point equidistant from two points?
No. Many points are, and together they form the perpendicular bisector.
How should the compasses be set to construct a perpendicular bisector?
To more than half the length of the segment.
What can you draw instead of arcs when space is limited?
Two isosceles triangles on the same base, then join the two apex points.
The angle bisector is the locus of which points?
Points equidistant from the two arms that meet at the vertex.
Angle bisector: what comes after the arc across both arms?
Equal-radius arcs from each crossing point that cross inside the angle, then join the vertex to that crossing.
How do you construct a perpendicular at a point on a line?
Cut the line either side of the point with equal arcs, then construct the perpendicular bisector of the part between the cut marks.
Where is the region closer to A than to B?
On A's side of the perpendicular bisector of AB. The bisector itself is the boundary.
With several conditions, what do you shade?
Only the region that satisfies all of the conditions, after constructing every boundary.
What is the locus of points a fixed distance from a point?
A circle centred on the point. Inside is closer, outside is further.
How is a bearing measured, and what does it give on its own?
Clockwise from North, in three figures. On its own it gives a ray (a locus), not a position.
How do you find the distance a boat has travelled?
Distance = speed × time.
How is the area found when a barrier is in the way?
Add the sectors of different radii around the corner. The answer can be left in terms of π.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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