INTRODUCTION TO COORDINATE GEOMETRY

     

Introduction to Coordinate Geometry

Number and Algebra : Module 29

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Years : 9-10

June 2011

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PDF Version of module


ASSUMED KNOWLEDGE

Fluency with the arithmetic of the rational numbersKnowledge of ratiosCongruent and similar trianglesBasic algebraic notationFluency with algebraic expressions và equationsBasic plotting points in the Cartesian plane including plotting points from a table of values.

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MOTIVATION

Coordinate geometry is one of the most important & exciting ideas of mathematics. In particular it is central to lớn the mathematics students meet at school. It provides a connection between algebra & geometry through graphs of lines & curves. This enables geometric problems khổng lồ be solved algebraically & provides geometric insights into algebra.

The invention of calculus was an extremely important development in mathematics that enabled mathematicians và physicists to model the real world in ways that was previously impossible. It brought together nearly all of algebra & geometry using the coordinate plane. The invention of calculus depended on the development of coordinate geometry.


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CONTENT

It is expected that students have met plotting points on the plane and have plotted points from tables of values of both linear và non linear functions.

The number plane (Cartesian plane) is divided into four quadrants by two perpendicular axes called the x-axis (horizontal line) & the y-axis (vertical line). These axes intersect at a point called the origin. The position of any point in the plane can be represented by an ordered pair of numbers (x, y). These ordered pairs are called the coordinates of the point.

*
The point with coordinates (4, 2) has been plotted on the Cartesian plane shown. The coordinates of the origin are (0, 0).

Once the coordinates of two points are known the distance between the two points và midpoint of the interval joining the points can be found.


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Distance between two points

Distances are always positive, or zero if the points coincide. The distance from A khổng lồ B is the same as the distance from B lớn A. We first find the distance between two points that are either vertically or horizontally aligned.


EXAMPLE

Find the distance between the following pairs of points.

a A(1, 2) and B(4, 2)b A(1, −2) và B(1, 3)

Solution

*
a The distance AB = 4 − 1 = 3

Note: The distance AB is obtained from the difference of the x-coordinates of the two points.

*
b The distance AB = 3 − (−2) = 5

Note: The distance AB is obtained from the difference of the y-coordinates of the two points.


The example above considered the special cases when the line interval AB is either horizontal or vertical. Pythagoras’ theorem is used to lớn calculate the distance between two points when the line interval between them is neither vertical nor horizontal.

The distance between the points A(1, 2) & B(4, 6) is calculated below.

*

AC = 4 − 1 = 3 & BC = 6 − 2 = 4.

By Pythagoras’ theorem,

AB2 = 32 + 42 = 25

And so AB = 5

The general case

We can obtain a formula for the length of any interval. Suppose that P(x1, y1) & Q(x2, y2) are two points.

*

Form the right-angled triangle PQX, where X is the point (x2, y1),

PX = x2 − x1 or x1 − x2 & QX = y2 − y1 or y1 − y2

depending on the positions of p. And Q.

By Pythagoras’ theorem:

PQ2 = PX2 + QX2

= (x2 − x1)2 + (y2 − y1)2

Therefore PQ = QP =

*

Note that (x2 − x1)2 is the same as (x1 − x1)2 & therefore it doesn’t matter whether we go from p to Q or from Q to phường − the result is the same.


EXAMPLE

Find the distance between the points A(−4, −3) & B(5, 7).

Solution

In this case, x1 = −4, x2 = 5, y1 = −3 và y2 = 7.

AB2 = (x2 − x1)2 + (y2 − y1)2

= (5 − (−4))2 + (7 − (−3))2

= 92 + 102

= 181

Thus, AB =

*

Note that we could have chosen x1 = 5, x2 = −4, y1 = 7 & y2 = −3 and still obtained the same result. As long as (x1, y1) refers to lớn one point & (x2, y2) the other point, it does not matter which one is which.


EXERCISE 1

Show that the distance between the points A(a, b) và B(c, d) is the same as the distance between

the points P(a, d) and Q(c, b) the points U(b, a) và V(d, c)

Illustrate both of these.


EXERCISE 2

The distance between the points (1, a) and (4, 8) is 5. Find the possible values of a và use a diagram to illustrate.


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The midpoint of an interval

The coordinates of the midpoint of a line interval can be found using averages as we will see.

We first deal with the situation where the points are horizontally or vertically aligned.


EXAMPLE

Find the coordinates of the midpoint of the line interval AB, given:

aA(1, 2) & B(7, 2) b A(1, −2) và B(1, 3)

Solution

*
a AB is a horizontal line interval, the midpoint is at (4, 2), since 4 is halfway between 1 và 7.

Note: 4 is the average of 1 và 7, that is, 4 =

*
.

*

b The midpoint of AB has coordinates 1,

*
.

note that

*
is the average of 3 và −2.


When the interval is not parallel to lớn one of the axes we take the average of the x-coordinate and the y-coordinate. This is proved below.

*

Let M be the midpoint of the line AB. Triangles AMS and MBT are congruent triangles (AAS), and so AS=MT and MS=BT.

Hence the x-coordinate of M is the average of 1 và 5.

x =

*
= 3

The y coordinate of M is the average of 2 & 8.

x =

*
= 5

Thus the coordinates of the midpoint M are (3, 5).

The general case

We can find a formula for the midpoint of any interval. Suppose that P(x1, y1) & Q(x2, y2)are two points và let M(x, y) be the midpoint.

*

Triangles PMS & MQT are congruent triangles (AAS), và so PS=MT & MS=QT.

Hence the x-coordinate of M is the average of x1 và x2, and y-coordinate of M is the average of y1 and y2. Therefore

x =

*
& y =
*

Midpoint of an interval

The midpoint of an interval with endpoints P(x1, y1) and Q(x2, y2) is

*
,
*
.

Take the average of the x-coordinates and the average of the y-coordinates.


EXAMPLE

Find the coordinates of the midpoint of the line interval joining the points (6, 8) và (−3, 2).

Solution

The midpoint has coordinates,

*
,
*
=
*
, 5


EXAMPLE

If C(3, 6) is the midpoint of line interval AB and A has coordinates (−1, 1), find the coordinates of B.

Solution

Let the coordinates of B be (x1, y1).

*
= 3 và
*
= 6

x1 − 1 = 6 y1 + 1 = 12

so x1 = 7 so y1 = 11.

Thus B has coordinates (7, 11).


EXERCISE 3

A square has vertices O(0, 0), A(a, 0), B(a, a) and C(0, a).

a Find the midpoint of the diagonals OB và CA. B Find the length of a diagonal of the square và the radius of the circle in which OABC is inscribed. C Find the equation of the circle inscribing the square.

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The gradient of a line

Gradient of an interval

The gradient is a measure of the steepness of line. There are several ways lớn measure steepness. In coordinate geometry the standard way to lớn define the gradient of an interval AB is

*
where rise is the change in the y-values as you move from A to B và run is the change in the x-values as you move from A to B. We will usually the pronumeral m for gradient.

*

Given the points A(2, 1) and B(5, 6):

gradient of interval AB =

*
=
*
=
*

Notice that as you move from A khổng lồ B along the interval the y-value increases as the x-value increases. The gradient is positive.

*
Given the points A(2, 7) và B(6, 1)

gradient of interval AB =

*
=
*
=
*
= −
*

or

gradient of interval bố =

*
=
*
= −
*

Notice that in this case as we move from A to lớn B the y value decreases as the x value increases. The gradient is negative. Similarly the gradient of tía = −

*
which is the same as the gradient of AB.

*
In general:

gradient of line interval AB =

*

m =

*

Note that since

*
=
*
it does not matter which point we take as the first & which point we take as the second.

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If the interval is vertical, the run is zero and the gradient of the interval is not defined. This is shown by interval AB.

If the interval is horizontal, the rise is zero as shown by interval PQ. The gradient of the interval is zero.

Gradient of a line

The gradient of a line is defined lớn be the gradient of any interval within the line.

This definition depends on the fact that two intervals on a line have the same gradient.

*

Suppose AB & PQ are two intervals on the same straight line. Draw right-angled triangles ABX & PQY with sides AX và PY parallel to lớn the x-axis và sides BX and QY parallel to the y-axis.

Triangle ABX is similar to triangle PQY since the corresponding angles are equal. Therefore:

*
=
*

That is, the intervals have the same gradient.


EXAMPLE

A line passes through the points (1,2) and (5,10). Find its gradient.

Solution

*
gradient =
*

=

*

= 2


EXAMPLE

A line passes through the point (5, 7) & has gradient

*
. Find the x-coordinate of a point on the line when y = 13.

Solution

*
Gradient of the line =
*
. Thus,

*
=
*

18 = 2(x − 5)

9 = x − 5

x = 14


EXERCISE 4

Find the gradient of the line passing through (a, b) and (0, c)


Intercepts

The x-intercept of a line is the point at which it crosses the x-axis.

The y-intercept of a line is the point at which it crosses the y-axis.

*

In the diagram khổng lồ the left the y-intercept is at A và the x-intercept at B.

The second diagram shows a line parallel khổng lồ the x-axis and it has a y-intercept at C.

The third diagram shows a line parallel to the y-axis and it has an x-intercept at D.


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Equation of a straight line

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When we plot points which satisfy the equation y = 2x + 1 we find that they lie in a straight line.

Can we find the equation of the line given suitable geometric information about the line? The following shows that this can be done given the gradient of the line and the y-intercept.


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The line y = 3x + 2

Consider the line with gradient 3 & y-intercept2. This passes through the point A(0, 2). Let B(x,y) be any point on this line.

*
Gradient of interval AB =
*
=
*
=
*

The gradient of the line is 3.

So,
*
= 3
Rearrangingy − 2 = 3x
y= 3x + 2

So the coordinates of B(x, y) satisfies y = 3x + 2. This is called the equation of the line.

Conversely suppose that B(x, y) satisfies the equation y = 3x + 2, then

*
= 3 and it passes the point (0, 2) so the point lies on the line with gradient 3 & y-intercept 2.

We summarise this by saying that the equation of the line is y = 3x + 2.


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The equation y = mx + c

Consider the line with gradient m and y-intercept c. If passes through the point A(0, c). Let B(x, y) be any point on this line.

*
Gradient of interval AB=
*
=
*

We know the gradient of the line is m.

Therefore
*
= m
y − c= mx
y= mx + c

That is, the line in the cartesian plane with gradient m and y-intercept c has equation y = mx + c. Conversely, the points whose coordinates satisfy the equation y = mx + c always lie on the line with gradient m and y-intercept c.

Vertical and horizontal lines

Vertical lines

In a vertical line all points have the same x-coordinate, but the y-coordinate can take any value. The equation of the vertical line through the point (6, 0) is x = 6. The x-axis intercept is 6. All the points on this line have x-coordinate 6.

*

In general, the equation of the vertical line through P(a,b) is x=a.

Because this line does not have a gradient it cannot be written in the form

y = mx + b.

Horizontal lines

A horizontal line has gradient 0. In a horizontal line all points have the same y-coordinate, but the x-coordinate can take any value. The equation of the horizontal line through the point (0, 5) is y = 5. The equation of the horizontal line through the point (9, 5) is y = 5.

*

In general, the equation of the horizontal line through P(a,b) is y=b.


EXAMPLE

Write down the gradient và y-intercept of the line with equation y = 3x − 4.

Solution

The gradient of the line is 3 & the y-intercept is −4.


Sometimes an equation needs to be rearranged before the gradient and y-intercept can be determined. Consider the following example.


EXAMPLE

Rewrite the equation 2x + 3y = 6 in the khung y = mx + c and hence find the value of the gradient and y-intercept.

Solution

2x + 3y = 6
so 3y = 6 − 2x
that is, y = 2 −
*
Thus y = −
*
+ 2

The gradient of the line is −

*
and the y-intercept is 2.

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Equation of a line given its gradient & a point on the line

*
We want to find the equation of the line with gradient m & which passes through the point P(x1, y1).

Let P(x, y) be any point with x ≠ x1 on the line passing through the point A(x1, y1) and let m be the gradient of this line.

Using gradient,m =
*
andy − y1 = m(x − x1)

This is the equation of the straight line with gradient m passing through the point A(x1, y1).


EXAMPLE

Find the equation of the line that passes through the point (−2, 3) with gradient −4.

Solution

The equation for this line is:

y − y1 = m(x − x1)
y − 3 = −4(x − (−2))
that is, y − 3 = −4x − 8
y = −4x − 5

Note that it is usual lớn give the answer in the khung y = mx + c

Equation of a straight line given two points

Given two points A(x1, y1) & B(x2, y2) the equation of the line passing through the two points can be found.

The gradient m of the line passing through A(x1, y1) and B(x2, y2) =

*
, x2 ≠ x1

Substituting into y − y1 = m(x − x1) gives

y − y1 =

*
(x − x1) , x2 ≠ x1


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Graphing straight lines

Using the equation khổng lồ sketch the line

If you are given an equation of a straight line & asked khổng lồ draw its graph all you need to vì chưng is find two points whose coordinates satisfy the equation and plot the points. There are two commonly used methods to lớn find two points.

Using the y-intercept & one other point

Using the y-intercept & a second point the equation can be found .


EXAMPLE

Draw the graph of y = 2x + 3.

Solution

*
The y-intercept is 3 & the gradient 2.

Substitute x = 1, so y = 5 giving the point (1, 5) lies on the line.

Plot the two points & draw the line through them.

This method does not work if the line is parallel to lớn the y-axis.


EXAMPLE

The gradient of a line is −6 và the y-intercept is 2. Find the equation of the line & sketch it.

Solution

Using the y = mx + c form for the equation of a straight line.

*
The equation of the line is y = −6x + 2.

The point (0, 2) lies on the line.

Substitute x = 1 in y = −6x + 2
= −6 × 1 + 2
= −4

The point (1, −4) lies on the line. The graph is shown.


x-intercept y-intercept method

In this method both intercepts are found.

The x-intercept is found by substituting y=0 andThe y-intercept is found by substituting x=0.

This method does not work if the line is parallel lớn an axis or passes through the origin.


EXAMPLE

Using the x-intercept y-intercept method sketch the graph of:

a y = 3x − 4 b 2x + 3y + 6 = 0

Solution

*

a When x = 0, y = −4
When y = 0, 3x − 4 = 0
3x = 4
x =
*

*

b When x = 0, 3y + 6 = 0
3y = −6
y = −2
When y = 0, 2x + 6 = 0
2x = −6
x = −3

The general form for the equation of a straight line

The equations y = 2x − 3, x = 6 and 2x − 3y = 6 can be written as −2x + y + 3, x − 6 = 0 and 2x − 3y − 6 = 0 respectively.

The general form for the equation of a line is ax + by + c = 0 where a, b and c are constants và a ≠ 0 or b ≠ 0. The equation of every line can be put in general form. The general size is not unique. The equation x + 2y +1 = 0 is the same straight line as 2x + 4y + 2 = 0.


EXERCISE 5

An equilateral triangle ABC has coordinates O(0, 0), B(a, 0) và C(c, d).

a Find c & d in terms of a by using the fact that OB = BC = CO.

b Find the equation of the lines OB, BC and CO.


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Parallel & perpendicular lines

Parallel lines

If two lines l1 & l2 are parallel then corresponding angles are equal.

Conversely, if corresponding angles are equal then the lines are parallel.

*

Theorem

Two lines are parallel if they have the same gradient và conversely, two lines with the same gradient are parallel.

Proof

In the diagram, two lines are drawn & the right-angled triangles PQX and ABY are added with QX=BY.

*

If the lines are parallel then PQX = ABY (corresponding angles).

The two triangles are congruent by the AAS test.

Therefore PX = PY &

*
=
*
.

That is, the gradients are equal.

Conversely. If the gradients are equal

*
=
*
.

Now QX=BY và therefore PX = AY.

Hence the triangles QPX và ABY are congruent by the SAS test.

Hence the corresponding angles PQX và ABY are equal & the lines are parallel.


EXAMPLE

Show that the line passing through the points A(6, 4) & B(7, 11) is parallel khổng lồ the line passing through P(0, 0) and Q(2, 14).

Solution

Gradient of AB =

*
= 7

Gradient of PQ =

*
= 7

The two lines have the same gradient & so are parallel.


EXAMPLE

Find the equation of the line that is parallel lớn the line y = −2x + 6 & passing through the point A(1, 10).

Solution

The gradient of the line y = −2x + 6 is −2.

Therefore the line through the point A(1, 10) parallel lớn y = −2x + 6 has equation:

y − y1 = m(x − x1)
y − 10 = −2(x − 1)
y = −2x + 12

Perpendicular lines

When we draw y = 3x & draw a line perpendicular to it passing through the origin then it is clear that y = −ax where a is a small positive number. We will show that the equation is y = −

*
x.

We are now going to show the surprising result that ‘if two lines are perpendicular then the product of their gradients is −1’ (or if one is vertical và the other horizontal). The converse is also true. That is, ‘ If the hàng hóa of the gradients of two lines is −1 then they are perpendicular’.

We first consider the case when both lines pass through the origin.

Draw two lines passing through the origin with one of the lines having positive gradient and the other negative gradient.

*

Form right-angled triangles OPQ & OAB with OQ = OB.

Gradient of the line OA=
*
Gradient of the line through OP=
*
The product=
*
×
*
=
*
×
*
(since OQ = OB)
=
*

If the lines are perpendicular, POQ = AOB.

*

Therefore OPQ OAB (AAS),

so PQ=AB & the product, −

*
, of the gradients is−1.

Conversely If the sản phẩm is −1, then AB=PQ,

so OAB OPQ (SAS).

Therefore POQ = AOB and so AOP = 90°.

We have now proved the result for lines through the origin.

If we are given two lines anywhere in the plane, we can draw lines through the origin parallel to lớn the given two lines. The gradient of each new line is the same as the gradient of the corresponding original line. So the result holds for lines that bởi not necessarily pass through the origin.


EXAMPLE

Show that the line through the points A(6, 0) and B(0, 12) is perpendicular lớn the line through P(8,10) & Q(4, 8).

Solution

Gradient of AB =
*
= −2
Gradient of PQ =
*
=
*
=
*
(Gradient of AB) × (gradient of PQ) = −2 ×
*
= −1

Hence the lines are perpendicular.


Parallel and perpendicular lines

If two non-vertical lines are parallel then they have the same gradient. Conversely if two non-vertical lines have the same gradient then they are parallel.If two non-vertical lines are perpendicular then the hàng hóa of their gradients is −1. Conversely if the product of the gradients of two lines is −1 then they are perpendicular.

EXAMPLE

Find the equation of the line which passes through the point (1, 3) and is perpendicular lớn the line whose equation is y = 2x + 1.

Solution

Gradient of the line y = 2x + 1 is 2.

Gradient of a line perpendicular to lớn this line is −

*
.

The required equation is y − 3 =
*
(x − 1)
2(y − 3) =−(x − 1)
2y + x=7

Thus the equation of the required line is 2y + x = 7.


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Proofs with coordinate geometry

Coordinate geometry can be used to lớn prove results in Euclidean Geometry. An important aspect of doing this is placing objects on the Cartesian plane in a way that minimises calculations.


EXAMPLE

Prove that the midpoints of a parallelogram bisect each other using coordinate geometry.

Solution

Let the coordinates of the vertices be O(0, 0), A(a, 0), B(a + c, d) và C(c, d).

There is no loss in generality in placing the vertices of the parallelogram on the Cartesian plane in this way.

The midpoint M of OB =
*
,
*
The midpoint N of AC =
*
,
*

M = N và so the midpoints coincided which means that the diagonals bisect each other.


EXAMPLE

Prove that the diagonals of a rhombus bisect each other at right angles using coordinate geometry.

Solution

Let the coordinates of the vertices be O(0, 0), A(a, 0), B(a + c, d) & C(c, d).

Because it is a rhombus all the sides are of equal length.

OA = AB = BC = co
Gradient of OB =
*
and,
Gradient of AC =
*

The sản phẩm of the gradients of the diagonals =

*
×
*
=
*
.

OA2 = a2 & by Pythagoras’ theorem, AB2 = c2 + d2

so a2 = c2 + d2.

Hence d2 = −(c2 − a2)

Thus the hàng hóa of the gradients of the diagonals = −1.


EXERCISE 6

In any triangle ABC prove that AB2 + AC2 = 2(AD2 + DC2)

Where D is the midpoint of BC.


EXERCISE 7

Prove that mix of points equidistant from two given points is a straight line.


HISTORY

There were three facets of the development of coordinate geometry.

The invention of a system of coordinatesThe recognition of the correspondence between geometry và algebraThe graphic representation of relations and functions

The Greek mathematician Menaechmus (380−320 BC) proved theorems by using a method that was very close to using coordinates & it has sometimes been maintained that he had introduced coordinate geometry.

Apollonius of Perga (262−190 BC) dealt with problems in a manner that may be called an coordinate geometry of one dimension; with the question of finding points on a line that were in a ratio lớn the others. The results và ideas of the ancient Greeks provided a motivation for the development of coordinate geometry.

Coordinate geometry has traditionally been attributed to lớn René Descartes (1599−1650) và Pierre de Fermat (1601−1665) who independently provided the beginning of the subject as we know it today.


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ANSWERS khổng lồ EXERCISES

EXERCISE 1

AB2 = PQ2= UV2 = (d − b)2 + (c − a)2

*

EXERCISE 2

a = 4 or a = 12

*

EXERCISE 3

a
*
,
*
bdiagonal =
*
a, radius =
*
cx −
*
2 + y −
*
2 =
*

EXERCISE 4

*

EXERCISE 5

a

c =

*
, d =
*

b

OA: y = 0, OC: y =

*
, OB: y =
*
(x − a)

EXERCISE 6

Place the triangle so D is at the origin. Then let the coordinates of B và C be (−a, 0) và (a, 0) respectively. Let the coordinates of A be (d, c).

AB2=c2 + (d + a)2 và AC2 = c2 + (d − a)2
SoAB2 + AC2=2c2 + 2d2 + 2a2
AD2=c2 + d2 và DC2 = a2
HenceAB2 + AC2=2(AD2 + DC2)

EXERCISE 7

Let P(x, y) be a point equidistant from A(a, b) & C(c, d)

PA = PC

2(c − a)x + 2(d − b)y = d2 + c2 − a2 − b2

EXERCISE 8

Let the coordinates of the vertices be O(0, 0), A(a, c), B(m, n) and D(b, 0)

Show that the midpoint of all the required line segments has coordinates

*
(m + b + a),
*
(n + c)

The Improving Mathematics Education in Schools (TIMES) Project 2009-2011 was funded by the Australian Government Department of Education, Employment và Workplace Relations.

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The views expressed here are those of the author & do not necessarily represent the views of the Australian Government Department of Education, Employment và Workplace Relations.

© The University of Melbourne on behalf of the International Centre of Excellence for Education in Mathematics (ICE-EM), the education division of the Australian Mathematical Sciences Institute (cameraminhtan.vn), 2010 (except where otherwise indicated). This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License. https://creativecommons.cameraminhtan.vn/licenses/by-nc-nd/3.0/