1. MODULE 3: GEOMETRY,... Coordinate Geometry Graphs Circular measure
Trigonometry
Vectors
F. Ndererehe, BA Educational Sciences - UNR, MA Educational Sciences - UNR

2. COORDINATE GEOMETRY
The position of a point in a plane can be given by an ordered pair of numbers, written as (x,y) and called ‘cartesian coordinates’(named after French Mathematician Rene Descartes). y x
F. Ndererehe, BA Educational Sciences - UNR, MA Educational Sciences - UNR

3. COORDINATE GEOMETRY ctd
The coordinates of the midpoint of two points A(x1,y1) and B(x2,y2) is (x1 + x2, y1 +y2 )
The distance between two points is given by (x2 – x1)2 + (y2 – y1)2
The gradient of a straight line through (x1,y1) and (x2, y2) is given by y2 – y1
x2 – x1
F. Ndererehe, BA Educational Sciences - UNR, MA Educational Sciences - UNR

4. COORDINATE GEOMETRY ctd
The gradient of any line parallel to the x-axis is zero
The gradient of any line parallel to the y-axis is undefined
Parallel lines have equal gradients
The product of gradients of perpendicular lines is – 1
F. Ndererehe, BA Educational Sciences - UNR, MA Educational Sciences - UNR

5. COORDINATE GEOMETRY ctd
The equation of a straight line through (x1, x2) is y – y1 = m(x – x1) where m is the gradient. This is known as one point, gradient form.
The equation can also be given by y = mx + c where m is the gradient and c the y-intercept. This is known as gradient-intercept form.
The coordinates of the point of intersection of two straight line is found by solving simultaneously their respective equations.
F. Ndererehe, BA Educational Sciences - UNR, MA Educational Sciences - UNR

6. COORDINATE GEOMETRY ctd
Work out the following:
Plot the points A(1,2); B(2,0), C(0,2), D(3,3).
State the coordinates of the midpoint of A(1,2) and E(3, - 4)
A circle has centre at (1,2). One point on its circumference is (-3, -1). Find its radius.
Which of the lines through the following pairs of points are parallel? (a) (-1, 3), (4,5); (b) (3, -2), (5,1); (c) (-4, -3), (1, -1); (d) (-7, 4), (2,4).
Which of the lines through the following pairs of points are perpendicular? (a) (-4, -2), (-1,0); (b) (-1,-4), (2, -8); (c) (0,-5), (4, -2); (d) (1, 2), (5, -4).
F. Ndererehe, BA Educational Sciences - UNR, MA Educational Sciences - UNR

7. COORDINATE GEOMETRY ctd
Find, in its simplest form, the equation of the line through (a) (2,3) with gradient 1; (b) through (0,1) and (-1, 3); (c) through (-3, -1) and perpendicular to a line with gradient – ½.
Find the equation of the line through the point of intersection of 2x + 3y = 5 and 3x – y = 2, and which is parallel to 4y – x = 14
Two points have coordinates A(1,3) and C(7,7). Find the equation of the perpendicular bisector of AC.
F. Ndererehe, BA Educational Sciences - UNR, MA Educational Sciences - UNR

8. GRAPHS
Graphs can represent various functions, i.e. linear (e.g. y = mx + c), quadratic (e.g. y = ax2 + bx + c), exponential (e.g. y = ax ), logarithmic (e.g. y = log x), trigonometric (e.g. y = sin x), etc.
To draw the graph, you need a table of values with sufficient selected values of x and corresponding values of y and then plotted in the cartesian plan and joined by a smooth line or curve accordingly (y= 2x ). X -3 -2 -1 1 2 3 4 5 Y
1/8 8 16 32
F. Ndererehe, BA Educational Sciences - UNR, MA Educational Sciences - UNR

9. GRAPHS ctd
A quadratic function can have its graph sketched without a full table of values:
Shape of the curve: a > 0 or a < 0?
Y-intercept: f(0)
Roots if any: f(x) = 0
Turning point: where x = -b/2a axis of symmetry
Minimum or maximum value of f(x): f(-b/2a).
Sine and cosine are periodic (period of 360); tangent has a period of 180.
F. Ndererehe, BA Educational Sciences - UNR, MA Educational Sciences - UNR

10. GRAPHS ctd Work out the following:
On a graph paper and taking an appropriate scale, draw the graphs of (a) y = 3x -2; (b) y = 2x2 + 3x – 4; (c) y = 5x ; (d) y = ln x; (e) y = cos x (for 0 x 720)
On any plain paper and showing all the steps, sketch the graph of y – 2x2 + 3x = 5
F. Ndererehe, BA Educational Sciences - UNR, MA Educational Sciences - UNR

11. CIRCULAR MEASURE
Parts of a circle include: chord, diameter, radius, circumference, arc, segment, sector and angle at the centre.
F. Ndererehe, BA Educational Sciences - UNR, MA Educational Sciences - UNR

12. CIRCULAR MEASURE Properties of a circle:
A perpendicular diameter bisects a chord
Perpendicular bisector of chord passes through centre
Equal chords are equidistant from the centre
F. Ndererehe, BA Educational Sciences - UNR, MA Educational Sciences - UNR

13. CIRCULAR MEASURE ctd Work out the following:
1. A chord AB of length 10cm is drawn in a circle centre O of radius 7cm. Find the distance of the chord from O.
2. ABC is a vertical section through a hemispherical bowl centre O and radius 15cm. It contains water whose maximum depth is 8cm. What is the width of the water surface?
F. Ndererehe, BA Educational Sciences - UNR, MA Educational Sciences - UNR

14. CIRCULAR MEASURE ctd Tangents
A tangent is perpendicular to the radius at the point of contact
The two tangents from an external point are equal in length
If two circles touch, the point of contact lies on the line joining the two centres and the circles have a common tangent
F. Ndererehe, BA Educational Sciences - UNR, MA Educational Sciences - UNR

15. CIRCULAR MEASURE ctd Work out the following:
In the diagram below, TA and TB are tangents to a circle centre O. TOC is a straight line and <ATB = 80. Calculate (i) <ABT, (ii) <AOB, (iii) <ACO.
F. Ndererehe, BA Educational Sciences - UNR, MA Educational Sciences - UNR

16. CIRCULAR MEASURE ctd Angles in a circle:
The angle at the centre = twice the angle at the circumference (subtended by the same arc)
The angle in a semicircle is (therefore) a right angle
Angles in the same segment are equal (as each is half the angle at the centre)
F. Ndererehe, BA Educational Sciences - UNR, MA Educational Sciences - UNR

17. CIRCULAR MEASURE ctd Work out the following:
Find angles x, y, w, z in diagram below and show that AC is the diameter.
A z w x B K
D 
40
y C  A
F. Ndererehe, BA Educational Sciences - UNR, MA Educational Sciences - UNR

18. CIRCULAR MEASURE ctd
Cyclic quadrilaterals, i.e. With each vertex lying on the circumference of a circle.
The opposite angles of a cyclic quadrilateral add up to 180 and, conversely, if a pair of opposite angles of a quadrilateral add up to 180, the quadrilateral is cyclic.
An exterior angle of a cyclic quadrilateral = the interior opposite angle
F. Ndererehe, BA Educational Sciences - UNR, MA Educational Sciences - UNR

19. CIRCULAR MEASURE ctd Work out the following:
In the diagram below, O is the centre of the circle and ABCD is a cyclic quadrilateral. <AOB = 50 and <ADC = 66. Calculate <BAC B A C D O
F. Ndererehe, BA Educational Sciences - UNR, MA Educational Sciences - UNR

20. CIRCULAR MEASURE ctd Tangents
A tangent is perpendicular to the radius at the point of contact
The two tangents from an external point are equal in length
If two circles touch, the point of contact lies on the line joining the two centres and the circles have a common tangent
F. Ndererehe, BA Educational Sciences - UNR, MA Educational Sciences - UNR

21. CIRCULAR MEASURE ctd Work out the following:
In the diagram below, TA and TB are tangents to a circle centre O. TOC is a straight line and <ATB = 80. Calculate (i) <ABT, (ii) <AOB, (iii) <ACO.
F. Ndererehe, BA Educational Sciences - UNR, MA Educational Sciences - UNR

22. TRIGONOMETRY ctd
For angles >90 we refer to a conventional circle of radius 1 unit and centre O in the cartesian plan Sin = y-coordinate, Cos = x-coordinate and Tan = y-coord/x-coord, i.e. Sin/Cos. The circle is divided in four quadrants.
Sine, cosine and tangent behave in different but specific ways in each of the quadrant when it comes to their sign: All three are positive in 1st Q, only Sine is positive in the 2nd Q, only Tangent is positive in the 3rd Q and only Cosine is positive in the 4th Q [ASITACO].
F. Ndererehe, BA Educational Sciences - UNR, MA Educational Sciences - UNR

23. TRIGONOMETRY ctd
Angles are usually measured starting from the positive x-axis anticlockwise except when dealing with bearings in which case they are measured starting from North clockwise.
Some useful trigonometric identities:
sin2 + cos2 = 1; tan = sin/cos; cosec = 1/sin; sec = 1/cos; cot = 1/tan or cos/sin; tan2 + 1 = sec2; cot2 + 1 = cosec2.
F. Ndererehe, BA Educational Sciences - UNR, MA Educational Sciences - UNR

24. TRIGONOMETRY ctd Some basic rules [well labelled triangle ABC]:
Sine rule: a/sin A = b/sin B = c/sin C
Cosine rule:
a2 = b2 + c2 – 2bc cos A; b2 = a2 + c2 – 2ac cos B; c2 = a2+ b2 – 2ab cos C
Area is given by:
½ ab sin C or ½ bc sin A or ½ ac sin B
Reminder:
Angles can be expressed in degrees or in radians. 1 rad = 360/2 and 1 = 2/360 rad. Calculator can also be used for direct answer.
F. Ndererehe, BA Educational Sciences - UNR, MA Educational Sciences - UNR

25. TRIGONOMETRY ctd Work out the following:
Convert 53 in radians and 3 radians in degrees, correct to 3 d.p.
A caresupporter starts off from A to visit B, a client 500m on a bearing of 045 from A. She then proceeds to client C who stays on a bearing of 315 from B. (a) how far is client C from the caresupporter’s place (A) if C is on a bearing of 335 from A? (b) What will be the total distance covered by the caresupporter by the time she comes back home (in A) assuming that she was moving in straight lines from A to B , from B to C and from C to A. (answers correct to the nearest metre).
Triangle PQR is such that PQ = 2cm; PR = 5cm and <P =40. (a) Find QR; (b) Find <QRP, (c) Find the area of triangle PQR (all answers correct to 1 d.p.).
Solve for 0   360: (a) 3 cot = 5 cos; (b) 10 Sin cos  = 13
(c) sin ( - 30) = 0.4; (d) 3 cosec2  + cot Sin2  = 5
5. Prove that cot/tan + 1 = cosec2 
F. Ndererehe, BA Educational Sciences - UNR, MA Educational Sciences - UNR

26. VECTORS
A vector is always characterised by its magnitude and direction, e.g. A force of 7N upwards
Vectors are represented by directed line segments and named after the segments’ ends(capital letters) or by single letters (small): AB, a, or a A > B
/AB/: Magnitude of vector AB, read ‘modulus AB’ and always positive. If A(X1, y1) and B(X2, y2) then
/AB/ = (x2 - x1)2 + (y2 – y1)2
Vectors that have the same direction and magnitude and are parallel are equal
A vector with no direction nor magnitude is a zero vector, denoted O
F. Ndererehe, BA Educational Sciences - UNR, MA Educational Sciences - UNR

27. VECTORS ctd
If a and b are parallel, then a = kb (with k0) and with same direction if k>0 and opposite directions if k< a
-3/2 a
1/2a -a
Note: scalar multiples of a vector can be combined arithmetically:
i.e. ½ a + ( - 3/2 a) = ( ½ - 3/2 )a = - a or ma + na = (m+n)a
F. Ndererehe, BA Educational Sciences - UNR, MA Educational Sciences - UNR

28. VECTORS ctd Work out the following:
1. On a graph paper draw vectors a with 2cm horizontally and 4 cm vertically, p with 12/5cm horizontally and 24/5cm vertically and in opposite direction with regard to a, q with 4/5cm horizontally and 8/5cm vertically with the same direction as a and r with 3cm horizontally and 6cm vertically in same direction as a. Then state each of the vectors p, q, and r in the form ka.
2. The line AB is divided into three equal parts at C and D. If AD = a, state as a scalar multiples of a, (a) AB, (b) CB, (c) BD
3. If P(-2, -5) and Q (3,7), find /PQ/.
F. Ndererehe, BA Educational Sciences - UNR, MA Educational Sciences - UNR

29. VECTORS ctd
Position vector: if O is the origin, then OA is called the position vector of A, e.g. If the position vector of A is 2a – 3b OA = 2a – 3b
Addition and subtraction of vectors:
Graphical method: “tail-to-head” and “parallelogram law”.
Analytical method: aka ”components” or “ij” method, where i is a vector of magnitude 1 unit in the direction of the x-axis and j is a vector of magnitude 1 in the direction of the y-axis. E.g. If AC has components AD (horizontal) and DC(vertical) of magnitude 3 and 4 parallel to the axes, AC = AD + DC = 3i + 4j and can be written as a column vector
F. Ndererehe, BA Educational Sciences - UNR, MA Educational Sciences - UNR

30. VECTORS ctd Some rules to remember:
(i) If ma + nb = pa + qb  m = p and n = q
(ii) If P, Q and R are collinear, then PQ = kQR or PQ = kPR
(iii) If ma + nb and pa + qb are parallel  m/p = n/q
Work out the following:
1.On a graph paper, draw vectors a with 4 units horizontally to the right and 2 units vertically upwards, vector c with 3units horizontally to the left and 2 units vertically upwards, vector b with 0 units horizontally and 3units vertically upwards. Then draw vectors (a) a +2b – c; (b) a – b + c; (c) ½ a + b – 2c
F. Ndererehe, BA Educational Sciences - UNR, MA Educational Sciences - UNR

31. VECTORS ctd Work out the following (ctd)
2. In triangle OAB, OA = a, OB = b, and M is the mid point of AB. State, in terms of a and b, (a) AB, (b) AM, (c) OM.
3. OA = p + q, OB = 2p – q; where p and q are two vectors and M is the mid-point of AB. Find, in terms of p and q (a) AB, (b) AM, (c) OM
4. Given that p = 3a – b and q = 2a – 3b, find the numbers x and y such that xp + yq = a + 9b
5. The position vectors of P, Q and R are 2a – b, (a – b), and a + b respectively. Find the value of  if PQR is a straight line. State the ratio PQ:QR
6. The position vectors of A and B are a and b respectively, relative to O. C lies on OB where OC:CB = 1:3. AC is produced to D where AD = pAC. If DB//OA, find the value of P.
F. Ndererehe, BA Educational Sciences - UNR, MA Educational Sciences - UNR

32. VECTORS ctd Work out the following ctd 7. B E O D A
Points A and B have position vectors a and b respectively relative to an origin O as shown on the diagram. The point D is such that OD = pOA and the point E is such AE = qAB. BD and OE intersect in X. if OX = 2/5OE and XB = 4/5DB, express OX and XB in terms of a, b, p and q and hence evaluate p and q X
F. Ndererehe, BA Educational Sciences - UNR, MA Educational Sciences - UNR

33. VECTORS ctd Work out the following(ctd)
8. A, B, C are points with position vectors 2i – 3j, i + 2j, 4i – j respectively. Find, in terms of i and j, the vectors AB, BC and CA.
9. Find the magnitude and the angle made with the positive x-axis of the vectors (a) ( 3), (b) -4i – 2j.
(-3)
10. The position vectors of A and B are 2i + 3j and 3i – 8j respectively. D is the mid-point of AB and E divides OD in the ratio 2:3. Find the coordinates of E.
11. The points A, B, C and D have position vectors i, 2i + 3j, 2i + j and 5i respectively. Show that AB and CD are perpendicular.
12. Find the magnitude and direction of the resultant of the following coplanar forces acting at point O: 10N in direction 000, 5N in direction 090, 20N in direction 135, 10N in direction 225.
13. Find the magnitude and direction of the resultant of forces 10, 20, 30 and 40N acting in directions 060, 120, 180, 270 respectively.
F. Ndererehe, BA Educational Sciences - UNR, MA Educational Sciences - UNR