Download IB Studies Geometry Package

IB Studies Geometry Package
Distance between two points ( x1 , y1 ) and ( x2 , y2 ) : d 
( x1  x 2 ) 2  ( y1  y 2 )
 x1  x2 y1  y 2 
,

2 
 2
Coordinates of the midpoint of a line segment with endpoints ( x1 , y1 ) and ( x2 , y2 ) : 
Equation of a straight line: y  mx  c
Gradient Formula:
1.
ax  bx  d  0
y 2  y1
x2  x1
In the diagram, the lines L1 and L2 are parallel.
y
6
4
2
–6
–4
–2
x
0
2
4
6
–2
–4
L1
–6
L2
(a)
What is the gradient of L1?
(b)
Write down the equation of L1.
(c)
Write down the equation of L2 in the form ax + by + c = 0.
(Total 4 marks)
2.
Three points are given A(0, 4), B(6, 0) and C(8, 3).
(a)
Calculate the gradient (slope) of line AB.
(2)
(b)
Find the coordinates of the midpoint, M, of the line AC.
(2)
(c)
Calculate the length of line AC.
(2)
(d)
Find the equation of the line BM giving your answer in the form ax + by + d = 0 where a, b and
d
.
(5)
(e)
State whether the line AB is perpendicular to the line BC showing clearly your working and
reasoning.
(3)
(Total 14 marks)
1
3.
The vertices of quadrilateral ABCD as shown in the diagram are A (–8, 8), B (8, 3), C (7,–1) and D (–
4, 1).
y
A
8
7
6
5
4
3
B
2
1
D
–8
–7
–6
–5
–4
–3
–2
–1
0
1
2
3
4
5
–1
6
7
8
x
C
–2
The gradient of the line AB is –
(a)
5
.
16
Calculate the gradient of the line DC.
(2)
(b)
State whether or not DC is parallel to AB and give a reason for your answer.
(2)
The equation of the line through A and C is 3x + 5y = 16.
(c)
Find the equation of the line through B and D expressing your answer in the form ax + by = c,
where a, b and c  .
(5)
The lines AC and BD intersect at point T.
(d)
Calculate the coordinates of T.
(4)
(Total 13 marks)
4.
The mid-point, M, of the line joining A(s, 8) to B(−2, t) has coordinates M(2, 3).
(a)
Calculate the values of s and t.
(b)
Find the equation of the straight line perpendicular to AB, passing through
the point M.
(2)
(4)
(Total 6 marks)
2
5.
The coordinates of the vertices of a triangle are P (–2, 6), Q (6, 2) and R (–8, a).
(a)
On graph paper, mark the points P and Q on a set of coordinate axes.
Use 1 cm to represent 1 unit on each axis.
(3)
(b)
(i)
Calculate the distance PQ.
(2)
(ii)
Find the gradient of the line PQ.
(3)
(iii)
If angle RPQ is a right angle, what is the gradient of the line PR?
(1)
(iv)
Use your answer from (b) (iii), or otherwise, to find the value of ‘a’.
(2)
(c)
The length of PR is 180 . Find the area of triangle PQR.
(2)
(Total 13 marks)
6.
The gradients of several lines are as follows:
Line
a
b
c
d
e
f
g
h
Gradient
−3
–5
2
1
3
0.5
3
6
–2
5
5
–2
0.4
(a)
Find two pairs of lines that are parallel to each other.
(b)
Find any two pairs of lines that are at right angles to each other.
(Total 4 marks)
7.
The equation of a line l1 is y =
(a)
1
x.
2
On the grid, draw and label the line l1.
y
9
8
7
6
5
4
3
2
1
0
0 1 2 3 4 5 6 7 8 9 x
The line l2 has the same gradient as l1, but crosses the y-axis at 3.
(b)
What is the geometric relationship between l1 and l2?
(c)
Write down the equation of l2.
(d)
On the same grid as in part (a), draw the line l2.
(Total 4 marks)
3
8.
Vanessa wants to rent a place for her wedding reception. She obtains two quotations.
(a)
The local council will charge her £30 for the use of the community hall plus £10 per guest.
(i)
Copy and complete this table for charges made by the local council.
Number of guests (N)
10
30
50
70
90
Charges (C) in £
(2)
(ii)
On graph paper, using suitable scales, draw and label a graph showing the charges. Take
the horizontal axis as the number of guests and the vertical axis as the charges.
(3)
(iii)
Write a formula for C, in terms N, that can be used by the local council to calculate their
charges.
(1)
(b)
The local hotel calculates charges for their conference room using the formula:
C=
5N
+ 500
2
where C is the charge in £ and N is the number of guests.
(i)
Describe, in words only, what this formula means.
(2)
(ii)
Copy and complete this table for the charges made by the hotel.
Number of guests (N)
0
20
40
80
Charges (C) in £
(2)
(iii)
On the same axes used in part (a)(ii), draw this graph of C. Label your graph clearly.
(2)
(c)
Explain, briefly, what the two graphs tell you about the charges made.
(2)
(d)
Using your graphs or otherwise, find
(i)
the cost of renting the community hall if there are 87 guests;
(2)
(ii)
the number of guests if the hotel charges £650;
(2)
(iii)
the difference in charges between the council and the hotel if there are 82 guests at the
reception.
(2)
(Total 20 marks)
4
9.
The costs charged by two taxi services are represented by the two parallel lines on the following
graph. The Speedy Taxi Service charges $1.80, plus 10 cents for each kilometre.
c
2.60
Speedy Taxi Service
2.40
2.20
2.00
cost ($)
1.80
1.60
Economic Taxi Service
1.40
1.20
1.00
0
2
4
6
8
distance (km)
k
(a)
Write an equation for the cost, c, in $, of using the Economic Taxi Service for any number of
kilometres, k.
(b)
Bruce uses the Economic Taxi Service.
(i)
How much will he pay for travelling 7 km?
(ii)
How far can he travel for $2.40?
(Total 4 marks)
10.
A is the point (2, 3), and B is the point (4, 9).
(a)
Find the gradient of the line segment [AB].
(b)
Find the gradient of a line perpendicular to the line segment [AB].
(c)
The line 2x + by – 12 = 0 is perpendicular to the line segment [AB]. What is the value of b?
(Total 4 marks)
5
11.
The following diagram shows the points P, Q and M. M is the midpoint of [PQ].
y
P(0, 2)
M
0
x
Q(2, 0)
(a)
Write down the equation of the line (PQ).
(b)
Write down the equation of the line through M which is perpendicular to the line (PQ).
(Total 4 marks)
12.
The following diagram shows a straight line l.
y
l
10
8
6
4
2
0
x
0
1
2
3
4
5
6
(a)
Find the equation of the line l.
(b)
The line n is parallel to l and passes through the point (0, 8). Write down the equation of the line
n.
(c)
The line n crosses the horizontal axis at the point P. Find the coordinates of P.
(Total 4 marks)
6