Faculty Faculty of Arts

Dean Prof. Michael Breen

Module Code MH 4722

Module Title Introduction to Geometry

Academic Year 2016/17

Programme and year of study BA 1

Semester Spring Semester 2016/17

Time allowed 2 hours

External Examiner Dr. Eabhnat Nı Fhloinn

Lecturer Dr. Bernd Kreussler

Percentage of total marks 60%

Authorised Materials Calculator, Geometry Maths Set, Mathematical Tables

Instructions to candidates Answer three of the following four questions.

Please do not turn over this page until you are instructed to do so.

Question 1:

(a) Draw a triangle ABC with the following approximate side lengths:2 marks

AB = 9, BC = 7 and CA = 5.

(b) Construct, with ruler and compass, the centroid of △ABC. You need to adhere8 marks

to the strict Euclidean Rules of construction in part (b).

(c) Write down a detailed description of your construction in part (b).10 marks

Question 2:

Let A = (51, 57), B = (74,−58) and C = (2,−62) be points in a Cartesian coordinatesystem in the plane.

(a) Determine the coordinates of the mid-point of the line segment AB.2 marks

(b) Determine an equation of the perpendicular bisector of the line segment AB.4 marks

(c) Determine an equation of the perpendicular bisector of the line segment BC.5 marks

(d) Find the coordinates of the circumcentre O of triangle ABC.5 marks

(e) Find the radius R of the circumcircle of triangle ABC.4 marks

Question 3:

Triangle ABC has sides b = 13, c = 7 and angle α = 37◦.We use standard notation as shown in the diagram.

b

BbC

bA

c b

a

β γ

α

Determine the following quantities:20 marks

(a) the side a;

(b) the angles β and γ;

(c) the radius R of the circumcircle of △ABC;

(d) the area of △ABC;

(e) the radius r of the incircle of △ABC.

Give your final results correctly rounded to two decimal places.

Question 4:

(a) The points A,B,C,D (in this order) are on a circle of which AC is a diameter. It10 marks

is known that AB = 2, BC = 9, CD = 6 and DA = 7.

(i) Find the exact value (surd form) of the length of the diameter AC.

(ii) Use addition theorems to find the exact values of sin(∠DCB) and cos(∠DCB).

(b) Draw a line segment AB of positive length.10 marks

Construct, with ruler and compass, a line segment of length5

7AB.

Make sure that all circles, arcs and line segments you construct are clearly visible.