Worksheets · Foundation and Higher

Compass constructions and loci

3 exam-style questions. Answers and the mark for each step are on the last page.

Question 1Non-calculator · 4 marks
Maya is bisecting angle $ABC$ with a ruler and compasses.
(a) Which is her first step? (1)
1. Join A to C and mark its midpoint
2. With the compass point on B, draw an arc crossing both arms
3. Measure the angle with a protractor and halve it
(b) Explain why the arcs drawn from the two points on the arms must have the same radius. (1)
(c) On paper, draw an angle of about $70^\circ$ and label it $ABC$ . Use a ruler and compasses to construct the bisector of angle $ABC$ . You must show all your construction arcs. Then write down, one step per line, what you drew, and check your drawing against the mark scheme. (2)
Question 2Non-calculator · 4 marks
A is $(-3,0)$ and B is $(3,0)$ . A point P is equidistant from A and B and is 4 units from the x-axis.
(a) Find both possible coordinates of P. (3)
(b) Describe the two loci whose intersections give the answers. (1)
Question 3Non-calculator · 5 marks
$A$ is the point $(-2, 1)$ and $B$ is the point $(4, 1)$ . The point $P$ lies on the perpendicular bisector of $AB$ . $P$ is also on the bisector of the angle between the positive $x$ -axis and the positive $y$ -axis.
(a) Find the coordinates of $P$ . (3)
(b) Work out the shortest distance from $P$ to the line $y = 7$ . (1)
(c) Explain why the shortest distance from $P$ to the line $y = 7$ is along a line perpendicular to $y = 7$ . (1)

Answers and marks

Question 1

(a) With the compass point on $B$ , draw an arc crossing both arms of the angle.

B1 The arc centred on $B$ .

(b) The crossing point must be the same distance from both points, so it lies on the line of symmetry of the angle.

C1 Linking equal radii to the crossing point being the same distance from both arm points (symmetry of the construction).

(c) An arc centred on $B$ crossing both arms at $X$ and $Y$ ; equal arcs from $X$ and $Y$ crossing at $Z$ ; the line $BZ$ , with every arc left visible.

B1 Arcs of equal radius drawn from two points on the arms that are the same distance from $B$ , crossing inside the angle.
B1 The bisector drawn from $B$ through the crossing point, within $2^\circ$ of the true bisector, with all construction arcs shown.

Question 2

(a) $(0,4)$ and $(0,-4)$

M1 The perpendicular bisector of AB is the y-axis.
M1 Being the given distance from the x-axis allows either sign of y.
A1 Correct answer: $(0,4)$ and $(0,-4)$

(b) The perpendicular bisector x = 0 and the two lines y = 4, y = -4.

C1 Correct conclusion with supporting reasoning: The perpendicular bisector x = 0 and the two lines y = 4, y = -4.

Question 3

(a) $(1, 1)$

P1 The perpendicular bisector $x = 1$ .
P1 The angle bisector $y = x$ .
A1 Correct answer: $(1, 1)$ .

(b) 6 units

B1 Correct answer: 6.

(c) Any other route is the hypotenuse of a right-angled triangle with the perpendicular as a shorter side.

C1 A correct argument: any slanted route is the hypotenuse of a right-angled triangle, so it is longer than the perpendicular.

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Compass constructions and loci

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