Unit 3 Coordinate Geometry Year 9 · 2012. 2. 24. · 5 Examples: 1. Find the equation of a line...
Transcript of Unit 3 Coordinate Geometry Year 9 · 2012. 2. 24. · 5 Examples: 1. Find the equation of a line...
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UNIT 3UNIT 3UNIT 3UNIT 3: : : : COORDINATE GEOMETRYCOORDINATE GEOMETRYCOORDINATE GEOMETRYCOORDINATE GEOMETRY
Unit 3.1Unit 3.1Unit 3.1Unit 3.1: : : : Formulae for Gradient, MidFormulae for Gradient, MidFormulae for Gradient, MidFormulae for Gradient, Mid----point & Distancepoint & Distancepoint & Distancepoint & Distance
(A) Gradient of a Straight Line
Examples:
1. Find the gradient of the line passing through A (-3, 2) and B (2, 3).
2. Find the gradient of the lines shown below:
(a)
(b)
3. Find the gradient of the line joining the points M (-3, 1) and
N (4, 7).
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(B) Midpoint of a Straight Line
Examples:
1. The diagram shows a line segment AB. Find the midpoint of line
AB.
2. Find the coordinates of the midpoint of the line segment joining
the following pair of points:
(a) A (3, 4) and B (5, 6) (b) P (-1, 2) and Q (3, - 4)
3. M (-1, 2) is the midpoint of the line segment joining the points A
and B. If the coordinates of A are (2, 3), find the coordinates of B.
4. The point M (s, 2) is the midpoint of the interval from (-3, -4) and
(5, t). Find the values of s and t.
5. A triangle has vertices A (1, 4), B (6, 0) and C (12, 4). Calculate
(a) the gradient of AB,
(b) the coordinates of the midpoint of BC.
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(C) Distance / Length of a Straight Line
Examples:
1. Calculate the distance between points U (2, 4) and V (-2, 1).
2. B is the point (2, 6) and C is the point (-3, 0). Calculate the length
of the line segment BC correct to 2 decimal places.
Unit 3.2Unit 3.2Unit 3.2Unit 3.2: : : : Gradient Gradient Gradient Gradient of Special Lines of Special Lines of Special Lines of Special Lines & Parallel Lines& Parallel Lines& Parallel Lines& Parallel Lines
(A) Gradient of a Horizontal Line
Gradient is always zero for a
horizontal line.
The equation of a
horizontal line is y = c
where c = y-intercept
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(B) Gradient of a Vertical Line
(C) Gradient of Parallel Lines
Gradient is always undefined for a
vertical line.
The equation of a vertical
line is x = k where k = x-
intercept
When two lines are parallel, they have
equation with the same gradient.
Equation of AB is 1y mx c= + and equation of PQ is
2y mx c= + where m is the gradient of the line.
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Examples:
1. Find the equation of a line which passes through the point (2, - 4)
and has a gradient of 0.
2. Find the equation of a line which passes through the point (3, - 1)
and has an undefined gradient.
3. Line PQ, is parallel to line AB. Given that line AB passes through
the points A (1, - 1) and B (2, 3), find the gradient of line PQ.
4. Line MN has a gradient of 5. Find the equation of another line PQ
passing through the origin and parallel to MN.
Classwork # ____ :
Gradient, Midpoint & Distance of a Straight Line Graph 1. Find the midpoint of the straight line joining the following pairs of
points:
(a) (2, 5) and (3, 7) (b) (- 2, - 1) and (- 4, - 7)
2. In each of the following, M is the midpoint of AB. Find the unknown
(a) M(3, 5), A(0, 4), B(x, y) (b) M(1, 4), A(- 2, a), B(b, 1)
3. For each of the pair of points below, find the length of the line segment
joining them. Give your answers correct to 2 decimal places.
(a) (2, 5) and (6, 10) (b) (- 2, - 3) and (4, 6)
4. Find the gradient and y-intercept of each of the following lines:
5. Line MN is parallel to the x – axis.
(a) What is the gradient of line MN?
(b) Write down the equation of line MN which passes through the
point (2, 4).
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Homework # ____ :
Gradient, Midpoint & Distance of a Straight Line Graph
1. Find the midpoint of the straight line joining the following pairs of
points:
(a) (- 3, - 4) and (4, 5) (b) (- 2, 5) and (6, - 3)
2. In each of the following, M is the midpoint of AB. Find the
unknown
(a) M(1, 3), A(- 1, 1), B(x, y) (b) M(8, 5), A(a, b), B(10, 2)
3. For each of the pair of points below, find the length of the line
segment joining them. Give your answers correct to 2 decimal
places.
(a) (0, - 1) and (5, - 2) (b) (- 3, - 2) and (- 1, 5)
Unit 3.3Unit 3.3Unit 3.3Unit 3.3: : : : Equation of a Straight LineEquation of a Straight LineEquation of a Straight LineEquation of a Straight Line
(A) Equation of a Straight Line given its Gradient and Y-intercept
Examples:
1. Write down the equation of the straight line given that
(a) the gradient is 3 and y-intercept is – 2 ,
(b) the gradient is 1
2− and y-intercept is 3.
2. Find the gradient and the y-intercept of the following straight
line equations:
(a) 2 4y x= + (b) 2
35
y x= −
(c) 2 6y x+ = (d) 2 5 15x y− − =
(e) 2 3 10 0y x− − = (f) 2 3 1
5 10 5x y− =
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(B) Equation of a Straight Line given its Gradient and a Point Examples:
1. Find the equation of line passing through the following point and
gradient
(a) 1
,2
5
; gradient = – 6. (b) (4, - 2); gradient = 3
4− .
2. Given that 3y x c= + passes through the point (1, 2), find c.
(C) Equation of a Straight Line given two points. Examples:
1. Find the equation of the straight line passing through the following
points:
(a) (4, 3) and (5, 5) (b) ( ),1 2− and (0, 5) (c) 1
2, 4
and 3
2,2
−
(D) Equation of a Straight Line given one point and the equation
of a parallel line. Examples:
1. Find the equation of the line parallel to 5 2 0y x+ + = and passing
through 1
,12
.
2. Find the equation of the line parallel to 3 4 3x y= − and passing
through (6,11) .
3. Find the equation of the line parallel to 8 0y + = and passing
through ( 3, 4)− .
4. Find the equation of the line parallel to 4 3 0x − = and passing
through (6,3) .
(E) Equation of a Straight Line of a given diagram. Examples:
Write down the equations of the lines AB, AC, AD, AE, BC and BE as
shown.
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Classwork # ____ : Equation of a Straight Line 1. Find the gradient and the y-intercept for each of the following
equations:
(a) 4y x= − + (b) 2
3y x= (c) 3 9 6 0y x− + =
2. Write down the equation of the line in general form, given
(a) gradient = – 3 and y-intercept = 2
(b) gradient = 1
4 and y-intercept =
2
3−
3. Find the equation of the straight line that passes through the
point (4, -2) with gradient 3
4− .
4. If 2y x c= + passes through a point (4, 1), find the value of c.
5. The equation of a straight line is 2 4x y+ = .
(a) Find the gradient of the line.
(b) Given that the point (5, k) lies on the line, find the value of k.
6. The straight line y mx c= + is parallel to the straight line
3 2y x= + and passes through the point (1, 2). Find the values of
m and c.
7. Find the equation of the line which passes through the points
(6, 5) and (4, 3).
8. Find the equation parallel to 2 3 0x y− + = and passing through
(5, 2).
Homework # ____ : Equation of a Straight Line 1. Find the gradient and the y-intercept for each of the following:
(a) 4 6x y− = (b) 4 2 3 0x y+ − =
2. Write down the equation of the line given its gradient is -1 and its
y-intercept is 3.
3. A straight line of gradient 3 passes through the point (0, 7). Write
down the equation of the line.
4. Find the equation of the line which passes through the points
(- 2, - 8) and (2, 4).
5. The line 2 3 0y x c+ − = passes through the point (- 1, 3). Find c.
6. Find the equation parallel to 2 3y x= − + and passing through
(1, 2).
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Unit 3.4Unit 3.4Unit 3.4Unit 3.4:::: Miscellaneous Problems on Coordinate GeometryMiscellaneous Problems on Coordinate GeometryMiscellaneous Problems on Coordinate GeometryMiscellaneous Problems on Coordinate Geometry
Examples:
1. The vertices of ∆PRT are P(0, 4), R(2, 5) and
T(4, - 4). TP is a line segment that is
perpendicular to RP.
Find the area of ∆PRT.
2. Three points A, B and C form an isosceles triangle where
AB = BC. Find
(a) the length of AB,
(b) the coordinates of C,
(c) the equation of the line
joining A and C.
3. ABCD is a trapezium which is symmetrical about the y-axis.
Given that 1
2AD BC= and the height of the trapezium is 4 units,
find
(a) the coordinates of D,
(b) the equation of the diagonal
joining B and D,
(c) the area of trapezium ABCD.
4. P is the point (4, 0), Q is the point (10, 4), R is the point (2, 6) and
O is the origin. Find
(a) PQ 2,
(b) the gradient of the line QR,
(c) the equation of the line through Q parallel to OR.
5. On the graph, O is the origin and l is the line which passes
through the points P(–2, 1) and R(4, 4). T is the point (1, 4).
(a) Find
(i) the gradient of l,
(ii) the equation of l,
(iii) the equation of the line through T parallel to l.
(b) (i) Write down the coordinates of M, the midpoint of PR.
(ii) Calculate the area of ∆TMR.
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Classwork # ____ : Problems on Coordinate Geometry
1. The vertices of a triangle are I(0, – 5), J(–9, 7) and K(16, 7).
Calculate
(a) the length of IJ,
(b) the length of JK,
(c) the length of KI,
(b) the perimeter of the triangle IJK.
2. In the diagram, the line BA meets the y-axis at C. Given that A is (2,
3) and B is (6, 5), calculate
(a) the coordinates of C,
(b) the gradient of AB,
(c) the length of AB,
(d) the area of trapezium ABNM.
3. A triangle has vertices A(1, 4), B(6, 0) and C(12, 4). Calculate
(a) the gradient of AB,
(b) the coordinate of the midpoint of BC.
4. The points A and B have coordinates (– 6, 2) and (6, 6)
respectively.
(a) Find the coordinates of the midpoint of AB.
(b) Calculate the length of AB.
(c) Find the gradient of the line AB.
(d) Find the equation of the line AB.
Homework # ____ : Problems on Coordinate Geometry
1. ABCD is a parallelogram.
(a) Find the coordinates of the midpoint of AC.
(b) Using the result in (a), find the coordinates of point D.
2. The line 2 6y x+ = cuts the x-axis at A and y-axis at B. The point
P is the midpoint of AB and O is the origin.
Find
(a) the coordinates of the point A,
(b) the coordinates of the point B,
(c) the length of AB,
(d) the equation of the line through P which is parallel to the
x-axis.