Basic Mathematics Form Four Notes – Coordinate Geometry

Equation of a Line

The General Equation of a Straight Line Derive the general equation of a straight line COORDINATES OF A POINT

The coordinates of a points – are the values of x and y enclosed by the brackets which are used

to describe the position of point in a line in the plane.

The plane is called xy-plane and it has two axis.

1. horizontal axis known as axis and

2. vertical axis known as axis

Consider the xy-plane below

The coordinates of points A, B, C ,D and E are A(2, 3), B(4, 4), C(-3, -1), D(2, -4) and E(1, 0).

Definition

change in y

© Consider two points A(x;,,) and B(x2, y2), the slope between the two points is given by:-

Yi — Y2 Xy—X2

(c) (—2,—3) and (—4,—7)

Y2—-1_—-7–3_-7+3_ -4 Xy—-X, —-4–2 -44+2 -2

(b) Find the value of m if the line joining the points (—5, —3) and (6, m) has a slope of i

Solution

5- -3

+. The value of k is —2

(b) Given (—5,—3) and (6,m) Yara

+. The value of k is

Exercise 1

1. Find the gradientof the line which passes through the following points ;

(3,6) and (-2,8)

b. (0,6) and (99,-12)

c. (4,5)and (5,4)

2. A line passes through (3, a) and (4, -2), what is the value of a if the slope of the line is 4? 3. The gradient of the linewhich goes through (4,3) and (-5,k) is 2. Find the value of k. FINDING THE EQUATION OF A STRAIGHT LINE

The equation of a straight line can be determined if one of the following is given:-

* Two points on the line

Example 3

c. Passing through the points and

Solution

y-N x—X y-4

Equation Of A Straight Line In Different Forms

The equation of a line can be expressed in two forms

Example 4

Intercepts

Example 5

Find the y-intercept of the following lines

Solution

Example 6

Find the x and y-intercept of the following lines

Exercise 2

Attempt the following Questions.

What is the x-intercept of the line passing through (3,3) and (-4,9)?

Find the equation of the straight line with a slope -4 and passing through the point (0,0).

Find the equation of the straight line with y-intercept 5 and passing through the point (- 4,8).

Graphs Of Straight Lines

The graph of straight line can be drawn by using the following methods; By using intercepts By using the table of values

Example 7

The coordinates are (5.0) and (0, -1)

Then show the straight line through the point (5 ' 0) and (0, -1) on the xy —plane.

By using the table of values

-1} 0 / 1 / 2{3 —3}-1/]1|3{5

Solving Simultaneous Equation By Graphical Method

. Use the intercepts to plot the straight lines of the simultaneous equations . The point where the two lines cross each other is the solution to the simultaneous

equations

Example 8

Solve the following simultaneous equations by graphical method

Draw a straight line of the point (2, 1.6) on the xy — plane

Draw a straight line of the point (—5.5, 3.7) on the xy — plane

Exercise 3 1. Draw the line 4x-2y=7 and 3x+y=7 on the same axis and hence determine their intersection point 2. Find the solutionfor each pair the following simultaneous equations by graphical method; y-x = 3 and 2x+y =9 3x- 4y=-1 and x+y = 2

Midpoint of a Line Segment The Coordinates of the Midpoint of a Line Segment

Determine the coordinates of the midpoint of a line segment Let S be a point with coordinates (x;,y:), T with coordinates (x2,y2) and M with coordinates (x,y)

where M is the mid-point of ST. Consider the figure below:

Tx2/¥2)

y2-y ¢ Changeiny M(x,y)

X2-X ion y-% S(x,y)

come x

T

Considering the angles of the triangles SMC and TMD, the triangles SMC and TMD are similar

since their equiangular

x-X, X2.-x

X2-X

SM __ y-Yi

Thus, the coordinates of M(x,y) are eS bi ws ans),

Generally, the coordinates of the mid-point of any line segment is given by ae a $i Li ws.

Example 9

Find the coordinates of the mid-point joining the points (-2,8) and (-4,-2)

Solution

The midpoint is given by ( a

The midpoint will be:

zz 2

Therefore the coordinates of the midpoint of the line joining the points (-2,8) and (-4, -2) is (- 3,3).

Distance Between Two Points on a Plane The Distance Between Two Points on a Plane

Calculate the distance between two points on a plane

Consider two points, A(x;,y:) and B(x2,y2) as shown in the figure below:

B(x2,¥2)

The distance between A and B in terms of x), yi,X2, and y2can be found as follows:Join AB and

draw doted lines as shown in the figure above.

Since the triangle ABC is a right angled, then by applying Pythagoras theorem to the triangle ABC we obtain

(Ab)? = (Ac)? + (Bc)?

Generally the distance between two points is given by:

y; be 7 and yp be -5 thus,

Therefore the distance is 13 units.

Parallel and Perpendicular Lines Gradients in order to Determine the Conditions for any Two Lines to be Parallel

Compute gradients in order to determine the conditions for any two lines to be parallel

The two lines which never meet when produced infinitely are called parallel lines. See figure

below:

Gradients in order to Determine the Conditions for any Two Lines to be Perpendicular

Compute gradients in order to determine the conditions for any two lines to be perpendicular

When two straight lines intersect at right angle, we say that the lines are perpendicular lines. See

an illustration below.

'P3(X3,¥3)

mr Qxs,¥2)

P,(x2,Y2)

Consider the points Pj(xi,yi), P2(x2,y2), P3(xs,y3), R(xi,y2) and Q(x3,y2) and the anglesa,p,y(alpha, beta and gamma respectively).

Therefore the triangle P2QP3is similar to triangle P;RP2

Ya Y2 × 3- y2~

ne i ee L

va 92 × 3— X2 _ ya7 Ya _ — _

X2— X4 1— X2

Example 10

Show that A(-3,1), B(1,2), C(0,-1) and D(-4,-2) are vertices of a parallelogram. Solution

Let us find the slope of the lines AB, DC, AD and BC

change iny

Thus, 2-1 1-(-3) 1—(-2)

2-(-1) _4 1-0

-1-(-2)_ 1 o-(-4) 4

We see that each two opposite sides of the parallelogram have equal slope. This means that the two opposite sides are parallel to each other, which is the distinctive feature of the parallelogram.

Therefore the given vertices are the vertices of a parallelogram.

Problems on Parallel and Perpendicular Lines

Solve problems on parallel and perpendicular lines Example 11 Show that A(-3,2), B(5,6) and C(7,2) are vertices of a right angled triangle.

Solution

Now,

Since the slope of AB and BC are negative reciprocals, then the triangle ABC is a right angled

triangle at B.