Loci Questions and Answers

Loci Questions

1. Using a ruler and a pair of compasses only,

1. Construct a triangle ABC such that angle ABC = 135^oC, AB = 8.2cm and BC = 9.6cm
2. Given that D is a position equidistant from both AB and BC and also from B and C
   1. Locate D
   2. Find the area of triangle DBC.

2. (a) Using a ruler, a pair of compasses only construct triangle XYZ such that XY = 6cm,

YZ = 8cm and XYZ = 75^o

 (b) Measure line XZ and XZY

 (c) Draw a circle that passes through X, Y and Z

(d) A point M moves such that it is always equidistant from Y and Z. construct the locus

of M and define the locus

3. (a) (i) Construct a triangle ABC in which AB=6cm, BC = 7cm and angle ABC = 75^o

Measure:-

(i) Length of AC

(ii) Angle ACB

 (b) Locus of P is such that BP = PC. Construct P

 (c) Construct the locus of Q such that Q is on one side of BC, opposite A and angle

BQC = 30^o

 (d) (i) Locus of P and locus of Q meet at X. Mark x

(ii) Construct locus R in which angle BRC 120^o

(iii) Show the locus S inside triangle ABC such that XS  SR

4 . Use a ruler and compasses only for all constructions in this question.

a) i) Construct a triangle ABC in which AB=8cm, and BC=7.5cm and ABC=112½°

ii) Measure the length of AC

 b) By shading the unwanted regions show the locus of P within the triangle ABC such that

i) AP ≤ BP

ii) AP >3cm

Mark the required region as P

 c) Construct a normal from C to meet AB produced at D

 d) Locate the locus of R in the same diagram such that the area of triangle ARB is ¾ the area

of the triangle ABC.

5. On a line AB which is 10 cm long and on the same side of the line, use a ruler and a pair of

compasses only to construct the following.

a) Triangle ABC whose area is 20 cm² and angle ACB = 90^o

b) (i) The locus of a point P such that angle APB = 45^o.

(ii) Locate the position of P such that triangle APB has a maximum area and calculate

this area.

6. A garden in the shape of a polygon with vertices A, B, C, D and E. AB = 2.5m, AE = 10m,

 ED = 5.2M and DC=6.9m. The bearing of B from A is 030º and A is due to east of E

 while D is due north of E, angle EDC = 110º,

 a) Using a scale of 1cm to represent 1m construct an accurate plan of the garden

 b) A foundation is to be placed near to CD than CB and no more than 6m from A,

i) Construct the locus of points equidistant from CB and CD.

ii) Construct the locus of points 6m from A

 c) i) shade and label R ,the region within which the foundation could be placed in the garden

ii) Construct the locus of points in the garden 3.4m from AE.

iii) Is it possible for the foundation to be 3.4m from AE and in the region?

7. a) Using a ruler and compasses only construct triangle PQR in which QR= 5cm, PR = 7cm

and angle PRQ = 135°

 b) Determine < PQR

 c) At P drop a perpendicular to meet QR produced at T d) Measure PT

 e) Locate a point A on TP produced such that the area of triangle AQR is equal to one-

and – a – half times the area of triangle PQR

 f) Complete triangle AQR and measure angle AQR

8. Use ruler and a pair of compasses only in this question.

 (a) Construct triangle ABC in which AB = 7 cm, BC = 8 cm and ∠ABC = 60⁰.

 (b) Measure (i) side AC (ii) ∠ ACB

 (c) Construct a circle passing through the three points A, B and C. Measure the radius of the

circle.

(d) Construct ∆ PBC such that P is on the same side of BC as point A and ∠ PCB = ½ ∠ ACB,

∠ BPC = ∠ BAC measure ∠ PBC.

9. Without using a set square or a protractor:-

(a) Construct triangle ABC in which BC is 6.7cm, angle ABC is 60^o and BAC is 90^o.

 (b) Mark point D on line BA produced such that line AD =3.5cm

 (c) Construct:-

 (i) A circle that touches lines AC and AD

(ii) A tangent to this circle parallel to line AD

. Use a pair of compasses and ruler only in this question;

 (a) Draw acute angled triangle ABC in which angle CAB = 37½ ^o, AB = 8cm and

CB = 5.4cm. Measure the length of side AC (hint 37½ ^o = ½ x 75^o)

 (b) On the triangle ABC below:

(i) On the same side of AC as B, draw the locus of a point X so that angle Ax C = 52½ ^o

(ii) Also draw the locus of another point Y, which is 6.8cm away from AC and on the

same side as X

 (c) Show by shading the region P outside the triangle such that angle APC  52 ½^o and

P is not less than 6.8cm away from AC

 Loci Answers

1.

(a)

(b) XYZ = 42^o
+ 1^o

XZ = 8.8 + 0.1 cm

c) Bisecting any two sides

Drawing the circle

(d)Perpendicular bisector of YZ

Identification of locus of M

2. AC = 8 cm  0.1

 ACB = 46⁰
 1⁰

3. a) AC = 12.9  0.1cm

b) i) Line and well shaded B2

c) h = 7  0.1

d)  ABC _____ Area = ½ x 8 x 7cm

 = 28cm

i.e. ¾ x 28 = Area for ARB

 = 21cm

i.e. ½ x 8 x h = 21

h = 5.25

4.

MP = 12 cm

 Area ∆ APB = ½ x 10 x 12 = 60 cm²

5. i)

ii) Yes

6. (a)

b) < PQR = 26° + 1°

d) 4.9 + 0.1cm

e) AT = u = 8.7cm

f)

7. a) ∆ ABC line AB = 7 cm and BC = 8 cm.

 Construction of ∡60^o

(b) AC = 7.6 + 0.1 and

∡ ACB = 53 + 1^o

(c) 2 sides bisector 1

Circle drawn radius 4.4. ± 0.1

(d) Bisect ∡ ACB

 Bisection line to cut the circle to identify P

∡ PBC measure ≡

(a) AB = 7 cm, BC = 8 cm

∡ ABC = 60^o

(b) AC = 7.6 + 0.1 cm

∡ ABC = 53^o±0.1

 (c) Perpendicular bisectors of any two sides.

Circle drawn

Radius = 4.4.±0.1. cm

(d) ∡ ACB bisected

Bisection line drawn to cut circle at P

∡BPC = ∡BAC = 67^o

∡
PBC = 88 ± 0.1^o

8. B1 – Line AC

B1Line AB

B1 AD

B3 – Drawing correct circle

B2- Tangent correctly drawn

9. a) B1 for constructing 150^o

B1 for constructing 75^o

B1 for completing triangle ABC

B1 for AC = 8.8 ± 0.1

(b) (i) B1For locating locus centre

B1 for locus of X

(ii) B1 for constructing arcs 6.8cm from AC

 B1 for locus Y

(c)B2 for shading the locus of P