Constructions
Level 9
Perpendicular & Angle Bisectors, Triangles, Quadrilaterals
Ability level: Basic
1) Construct a perpendicular bisector for each of the line segment.
(a)
PQ
(b) ST
2) Construct an angle bisector for each of the angle.
P
(a)
(b)
Q
R
T
S
Q
P
T
V
3) (a) Draw the diagram below using a ruler and a protractor only.
A
8 cm
60°
B
10 cm
C
(b) Construct the angle bisector of
using a ruler and compasses only.
(c) Mark a point T on the angle bisector which is 6 cm from B.
(d) (i) Draw the perpendicular line TS from T to meet AB at S using a set square only.
(ii) Measure and write down the length of ST.
U
Constructions
Level 9
Perpendicular & Angle Bisectors, Triangles, Quadrilaterals
4) (a) Using a ruler, protractor and compasses only, construct a quadrilateral PQRS in
which PQ = 10 cm, QR = 8 cm, PR = 13 cm,
and
.
(b) Measure
i.
The length of SQ correct to the nearest 0.1 cm,
ii.
The size of the largest angle of ∆PQR correct to the nearest degree.
5) In the figure below, CN is the perpendicular bisector of the line segment AB. Given that
AB = (5x – 3) cm, AC = (3x + 1) cm, BC = (4x – 2) cm and CN = (3x – 1) cm, find
(a)
the value of x,
(b)
the length CN,
(c)
the length AN.
6) In the figure below, AB is the angle bisector of ∠CAD. It is given that AC ⊥ BC, AD ⊥ BD,
∠BAD = x°, AB = (3y + 1) cm, BC = (y + 1) cm and BD = (3y – 7) cm. Find
(a)
the value of y,
(b)
the length AB,
(c)
the value of x if ∠CAD = 58°.
Constructions
Level 9
Perpendicular & Angle Bisectors, Triangles, Quadrilaterals
Ability level: Intermediate
7) (a) Draw the given figure using a ruler and a protractor.
(b) Using a ruler and compasses only, construct the angle bisector of
and
.
Label the point where the two bisectors meet as X.
(c) (i) Find
.
(ii) Hence, write down a special angle property of ∆BXC.
8) (a) Draw a line segment, AB 10 cm long.
(b)
Construct the perpendicular bisector of AB and label the intersection of the
perpendicular bisector with AB as M.
(c)
Mark a point C on the perpendicular bisector 5.4 cm above M.
(d)
Using a ruler, measure AC and BC. What do you think is the relationship between
AC and BC?
(e)
Using a protractor, measure ∠ACM and ∠BCM. What do you think is the
relationship between ∠ACM and ∠BCM?
Constructions
Level 9
Perpendicular & Angle Bisectors, Triangles, Quadrilaterals
9) (a) Draw and label triangle ABC such that AB = 7.9 cm, ∠BAC = 48° and AC = 4.8 cm.
(b)
Measure and write down the length of BC.
(c)
Measure and write down the size of ∠ACB .
(d)
Label the point, X, which is equidistant from the vertices of the triangle.
(e)
Measure and write down the length of AX.
10) Draw a quadrilateral ABCD given that AB = 7.7 cm, AD = 3.5 cm, ∠ADC = 105°,
∠BAD = 70° and ∠ABC = 40°.
(a)
Construct the angle bisector of ∠BAD .
(b)
Construct the perpendicular bisector of DC.
(c)
Label the point X where the angle bisector and the perpendicular bisector meet.
(d)
Construct a circle with centre X which passes through points D and C.
(e)
Write down the radius of the circle.
11) (a) Using a ruler and a protractor only, draw a triangle PQR with PQ = 7 cm, QR = 8 cm
and
.
(b) Using a ruler and compasses only, construct
I.
The perpendicular bisector of PQ,
II.
The angle bisector of
.
(c) Mark clearly a possible point inside ∆PQR which is equidistant from P and Q and is
nearer to QR than to QP. Label this point as T.
Constructions
Level 9
Perpendicular & Angle Bisectors, Triangles, Quadrilaterals
12) (a) (i) Draw a horizontal line ST 8.4 cm long.
(ii) Using a ruler and a protractor only, construct above ST, the triangle STU in which
and
.
(b) (i) Using a ruler and compasses only, construct the angle bisector of
.
(ii) Label the point where the angle bisector meets UT as P.
(c) Measure the length of SP and give your answer correct to the nearest 0.1 cm.
Ability level: Advanced
13) Using a ruler and compasses only,
(a) Draw a triangle XYZ with XY = 10 cm, XZ = 9 cm and YZ = 8 cm.
(b) Construct the perpendicular bisectors of XY, XZ and YZ and mark a point P which
lies on all the 3 perpendicular bisectors of XY, XZ and YZ.
14) (a) Draw the perpendicular bisectors of the sides of each of the following triangles using
a straight edge and compasses only.
(i)
(ii)
Constructions
Level 9
Perpendicular & Angle Bisectors, Triangles, Quadrilaterals
(iii)
(b) What do you observe from the constructions in (a)?
15) (a) (i) Draw a horizontal line PQ 14 cm long.
(ii) Construct above PQ, a quadrilateral PQRS in which
and
, QR = 6 cm,
.
(b) Measure
(i) the size of
and give your answer correct to the nearest degree,
(ii) the length of QS and give your answer correct to the nearest 0.1 cm.
(c) Construct the angle bisector of
(d) The angle bisector of
.
and the line segment QS meet at X.
(i) Label the point X.
(ii) Measure the length of PX and give your answer correct to the nearest 0.1 cm.
16) Using a ruler and compasses only, construct the quadrilateral MNPQ in which
MN = 8 cm,
, MQ = 4 cm, P is equidistant from M and N, and
.
a) Show with reasons that
i.
∆MNP is equilateral,
ii.
∆QMP is a right-angled triangle.
b) Measure the perpendicular length from P to MN and give your answer correct to
the nearest 0.1 cm.