The length of a line segment is the distance between its endpoints. For a line segment AB with endpoints A(x1,y1) and B(x2,y2), the run x2− x1 measures the distance between x1 and x2 and the rise y2− y1 measures the distance between y1 and y2.
B (x2,y2)
A (x1,y1)
x2–x1
y2–y1
Using Pythagoras’ theorem, (AB)2= (x
2 − x1)2 + (y2− y1)2.
This could be expressed as the distance-between-two-points formula:
d(A,B) = (x2− x1)2 + (y2 − y1)2, where d(A,B) is a symbol for the distance between the points A and B.
Coordinate geometry of the straight line
1 WE14 Show that the points A(−3,−12), B(0,3) and C(4,23) are collinear.
2 For the points P(−6,−8), Q(6,4) and R(−32,34), fi nd the equation of the line through P and Q and hence determine if the three points are collinear.
The length of the line segment is AB= (x2 − x1)2+ (y2 − y1)2.
ExErcisE 1.6 PractisE
Work without cas Questions 1–2
Calculate the length of the line segment joining the points A(−2,−5) and B(1,3).
tHinK WritE
1 Write the distance formula. d(A,B) = (x2 − x1)2+ (y2 − y1)2 2 Substitute the coordinates of the two points.
Note: It does not matter which point is labelled (x1,y1) and which (x2,y2). A(−2,−5) and B(1,3) Let A be (x1,y1) and B be (x2,y2). d(A,B) = (1 − (−2))2 + (3 − (−5))2 = (3)2 + (8)2 = 9 + 64 = 73 3 State the answer. By choice, both the exact surd value and
its approximate value to 2 decimal places have been given.
Note: Always re-read the question to see if the degree of accuracy is specifi ed.
Therefore the length of AB is 73 ≃ 8.54 units.
WOrKeD eXaMpLe
3 a WE15 Calculate, correct to 2 decimal places, the angle made with the positive direction
of the x-axis by the line which passes through the points (1,−8) and (5,−2).
b Calculate the angle of inclination with the horizontal made by a line which has a gradient of −2.
c Obtain the equation of the line which passes through the point (2,7) at an angle of 135o to the horizontal.
4 Calculate the angle of inclination with the horizontal made by each of the lines whose gradients are 5 and 4 respectively, and hence find the magnitude of the acute angle between these two lines.
5 a WE16 State the gradient of a line which is:
i parallel to 3y − 6x = 1
ii perpendicular to 3y − 6x = 1.
b Show that the lines y= x and y = −x are perpendicular.
c Determine the equation of the line through the point (1,1) perpendicular to the line y = 5x + 10.
6 Find the coordinates of the x-intercept of the line which passes through the point (8,−2), and is parallel to the line 2y − 4x = 7.
7 WE17 Calculate the coordinates of the midpoint of the line segment joining the
points (12,5) and (−9,−1).
8 If the midpoint of PQ has coordinates (3,0) and Q is the point (−10,10), find the coordinates of point P.
9 WE18 Determine the equation of the perpendicular bisector of the line segment
joining the points A(−4,4) and B(−3,10).
10 Given that the line ax + by = c is the perpendicular bisector of the line segment CD where C is the point (−2,−5) and D is the point (2,5), find the smallest non-negative values possible for the integers a, b and c.
11 WE19 Calculate the length of the line segment joining the points (6,−8) and (−4,−5).
12 Calculate the distance between the point (3,10) and the midpoint of the line segment AB where A is the point (−1,1) and B is the point (6,−1). Give the answer correct to 2 decimal places.
13 Calculate the magnitude of the angle the following lines make with the positive direction of the x-axis, expressing your answer correct to 2 decimal places where appropriate.
a b The line that cuts the x-axis at x = 4 and
the y-axis at y = 3
c The line that is parallel to the y axis and passes through the point (1,5)
d The line with gradient −7
consolidatE
apply the most appropriate mathematical
processes and tools (3, 9)
(–2, 0) 0
y
x θ
14 Determine the equation of the line, in the form ax+ by = c, which:
a passes through the point 0,6 and is parallel to the line 7y − 5x = 0
b passes through the point −2,4
5 and is parallel to the line 3y+ 4x = 2
c passes through the point −3
4,1 and is perpendicular to the line 2x− 3y + 7 = 0
d passes through the point (0,0) and is perpendicular to the line 3x − y = 2
e passes through the point (−6,12) making an angle of tan−1(1.5) with the horizontal
f passes through the point of intersection of the lines 2x− 3y = 18 and 5x+ y = 11, and is perpendicular to the line y = 8.
15 Given the points A(−7,2) and B(−13,10), obtain:
a the distance between the points A and B
b the coordinates of the midpoint of the line segment AB
c the equation of the perpendicular bisector of AB
d the coordinates of the point where the perpendicular bisector meets the line 3x + 4y = 24.
16 A line L cuts the x-axis at the point A where x = 4, and is inclined at an angle of 123.69° to the positive direction of the x-axis.
a Form the equation of the line L specifying its gradient to 1 decimal place.
b Form the equation of a second line, K, which passes through the same point A at right angles to the line L.
c What is the distance between the y-intercepts of K and L?
17 a Determine whether the points A(−4,13), B(7,−9) and C(12,−19) are collinear.
b Explain whether or not the points A(−15,−95), B(12,40) and C(20,75) may be joined to form a triangle.
c Given the points A(3,0), B(9,4), C(5,6) and D(−1,2), show that AC and BD bisect each other.
d Given the points P(−2,−3), Q(2,5), R(6,9) and S(2,1), show that PQRS is a parallelogram. Is PQRS a rectangle?
18 Triangle CDE has vertices C(−8,5), D(2,4) and E(0.4,0.8).
a Calculate its perimeter to the nearest whole number.
b Show that the magnitude of angle CED is 90°.
c Find the coordinates of M, the midpoint of its hypotenuse.
d Show that M is equidistant from each of the vertices of the triangle.
19 A circle has its centre at (4,8) and one end of the diameter at (−2, −2).
a Specify the coordinates of the other end of the diameter.
b Calculate the area of the circle as a multiple of π.
20 a Find the value of a so that the line ax− 7y = 8 is:
i parallel to the line 3y+ 6x = 7
ii perpendicular to the line 3y+ 6x = 7.
b Find the value of b if the three points (3,b), (4,2b) and (8,5− b) are collinear.
c Find the value of c if the line through the points (2c,−c) and (c,−c − 2) makes an angle of 45° with the horizontal.
d Find the value of d so the line containing the points (d+ 1,d− 1) and (4,8) is:
i parallel to the line which cuts the x-axis at x = 7 and the y-axis at y = −2
ii parallel to the x-axis
iii perpendicular to the x-axis.
e If the distance between the points (p,8) and (0,−4) is 13 units, find two possible values for p.
f The angle between the two lines with gradients −1.25 and 0.8 respectively has the magnitude α°. Calculate the value of α.
21 Two friends planning to spend some time bushwalking argue over which one of them should carry a rather heavy rucksack containing food and first aid items. Neither is keen so they agree to each throw a small coin towards the base of a tree and the person whose coin lands the greater distance from the
tree will have to carry the rucksack. Taking the tree as the origin, and the
distances in centimetres east and north of the origin as (x,y) coordinates, Anna’s coin lands at (−2.3,1.5) and Liam’s coin lands at (1.7,2.1). Who carries the rucksack, Anna or Liam? Support your answer with a mathematical argument.
22 The diagram shows a main highway through a country town. The section of this highway running between a petrol station at P and a restaurant at R can be considered a straight line. Relative to a fixed origin, the coordinates of the petrol station and restaurant are P(3,7) and R(5,3) respectively. Distances are measured in kilometres.
a How far apart are the petrol station and restaurant? Answer to 1 decimal place.
b Form the equation of the straight line PR.
Ada is running late for her waitressing job at the restaurant. She is still at home at the point H(2,3.5).
There is no direct route to the restaurant from her home, but there is a bicycle track that goes straight to the nearest point B on the highway from her home. Ada decides to ride her bike to point B and then to travel along the highway from B to the restaurant.
c Form the equation of the line through H perpendicular to PR.
Petrol station P(3, 7) B H(2, 3.5) Restaurant R(5, 3) 2 1 0 3 5 6 Highway North 3 4 5 6 1 2 7 East 4
d Hence find the coordinates of B, the closest point on the highway from her home.
e If Ada’s average speed is 10 km/h, how long, to the nearest minute, does it take her to reach the restaurant from her home?
For questions 23a and 24a, use the geometry facility on CAS technology to draw a triangle.
23 a Construct the perpendicular bisectors of each of the three sides of the triangle. What do you notice? Repeat this procedure using other triangles. Does your observation appear to hold for these triangles?
b For the triangle formed by joining the points O(0,0),A(6,0),B(4,4), find the point of intersection of the perpendicular bisectors of each side. Check your answer algebraically.
24 a Construct the line segments joining each vertex to the midpoint of the opposite side (these are called medians). What do you notice? Repeat this procedure using other triangles. Does your observation appear to hold for these triangles? b For the triangle formed by joining the points O(0,0),A(6,0)and B(4,4), find the point of intersection of the medians drawn to each side. Check your answer algebraically.
rené descartes, seventeenth century French mathematician and philosopher, was one of the first to combine algebra and geometry together as coordinate geometry in his work La Géométrie. MastEr
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