Mathematics 3204/05
Public Exam Questions
Formulae and Proofs
Distance/Midpoint/Slope
Formulae
Coordinate Geometry
Proofs
m
y 2  y1
x 2  x1
 x1  x 2 y1  y 2 
midpo int  
,

2 
 2
dis tan ce 
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 y 2  y1 2  x 2  x1 2
2007-2008
Mathematics 3204/05
1.
2.
Public Exam Questions
Formulae/Proofs
If a circle has a diameter with endpoints ( 2, 7) and (10,13) , what is the length of
the radius?
(A)
3 5
2
(B)
(C)
(D)
3 5
6 5
2 29
What is the distance between R and the midpoint of PQ ?
y
(A)
(B)
(C)
(D)
2
R (2, 4)
2
10
10
S
Q (2, 2)
P
x
3.
The endpoints of a diameter of a circle are
of the circle?
(A)
(B)
(C)
(D)
4.
 2, 5 and  6,1 . What is the radius
2
2 3
2 5
4 5
The endpoints of a diameter of a circle are P  2, 7  and Q  6, 5 . What is the
centre of the circle?
(A)
(B)
(C)
(D)
 8, 2
 2, 6
 4, 12 
 6, 2
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5.
(B)
(C)
(D)
(B)
(C)
(D)
 9, 7 
 4, 5
 4,  5
9,  7 
The endpoints of a diameter of a circle are A  3,  5 and B 1,  1 . What is the
length of the radius?
(A)
(B)
(C)
(D)
8.
 14, 14
 5,1
 2, 14 
 4, 16 
If AB and CD are perpendicular bisectors of each other, and CD has endpoints
C  5,  2  and D  13,12 , what is the point of intersection?
(A)
7.
Formulae/Proofs
A circle with centre  3, 5 has diameter AB with A  8,  4  . What are the
coordinates of B?
(A)
6.
Public Exam Questions
5
10
2 5
2 10
The line containing the points A  8, m  and B  2,1 is parallel to the line
containing points C 11,  1 and D  7,  3 . What is the value of ‘m’?
(A)
(B)
(C)
(D)
9
4
4
21
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279
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Mathematics 3204/05
9.
Formulae/Proofs
A circle with centre C (1, 6) has a diameter with endpoint (3, 9). What are the
coordinates of the other endpoint of this diameter?
(A)
(B)
(C)
(D)
10.
Public Exam Questions
(1, 3)
(2, 7.5)
(5, 12)
(9, 7)
If the diameter of a circle has endpoints of (4, 5) and  2,3 , what is the exact
length of the radius?
(A)
(B)
10
2 10
(C)
(D)
17
2 17
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Public Exam Questions
Formulae/Proofs
Answers Formulas
1. B
2. C
3. C
4. B
5. C
6. B
7. A
8. B
9. A
10. A
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Public Exam Questions
Formulae/Proofs
1. A quadrilateral has coordinates M (2, 6) , A (2, 4) , T ( 4, 4) and H ( 4, 6) . Sketch
the quadrilateral and prove that the diagonals are congruent.
2. A circle, centred at the origin, contains chord AB with endpoints
A  8,6 and B  6,8 . Find the perpendicular distance from the centre of the
circle to the chord.
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Public Exam Questions
Formulae/Proofs
3. A rectangle has vertices A(8, 0), B(2, 9), C(1, 7), and D(5, 2). Sketch the rectangle
and find its area.
4. A triangle has vertices P  2, 5 , Q  2, 4 , and R  4, 1 . Show that the triangle is
scalene.
5. Given the diagram as shown, use coordinate geometry to prove that the midpoint of
the hypotenuse, M, is an equal distance from A, B, and C.
y
A (0, 8)
M
C (0, 0)
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283
B (6, 0)
2007-2008
x
Mathematics 3204/05
Public Exam Questions
Formulae/Proofs
6. A circle with centre at the origin contains a chord having endpoints T  4, 2 and
Q  2,  4 . Using coordinate geometry, prove that the segment joining the midpoint of
the chord to the centre of the circle is perpendicular to the chord.
y
T (-4, 2)
O (0, 0)
x
Q (-2, -4)
7. Given ABC as shown, use coordinate geometry to prove BD is the perpendicular
bisector of AC .
y
B (6, 8)
A (-2, 2)
D (5, 1)
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284
C (12, 0)
x
2007-2008
Mathematics 3204/05
Public Exam Questions
Formulae/Proofs
8. In ABC , M and N are the midpoints of AC and BC respectively. Prove that the
length of MN is half the length of AB .
6– 5
1
2
3
4
5
1
2
3
4
1– 8
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
y
6
C (-4, 5)
5
4
B (8, 3)
3
2
1
– 8– 7– 6– 5– 4– 3– 2– 1
– 1
1
2
3
4
5
6
7
8
9
10
x
– 2
– 3
A (2, -3)
– 4
– 5
9. Given the square ABCD with vertices A  1,  1 , B  3, 2  , C  6,  2  and
D  2,  5 , prove that the diagonals are congruent.
y
B (3,2)
x
A (-1,-1)
C (6,-2)
D (2,-5)
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Public Exam Questions
Formulae/Proofs
10. If the points A 1, 1 , B  4,3 , and C  3, 2  are the vertices of a triangle, prove that
tthe triangle is isosceles.
6– 5
1
2
3
4
5
1
2
3
4
1– 8
2
3
4
5
6
7
8
9
10
1
2
3
4
5
6
7
y
6
5
4
B (4, 3)
3
C (-3, 2)
2
1
– 8– 7– 6– 5– 4– 3– 2– 1
– 1
– 2
1
2
3
4
5
6
7
8
9
10
x
A (1, -1)
– 3
– 4
– 5
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286
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Public Exam Questions
Formulae/Proofs
Answers
1.
y
10
8
6
T(-4, 4)
A(2, 4)
4
2
- 10
-8
-6
-4
-2
2
-2
4
6
8
10
x
-4
H(-4, -6)
M(2, -6)
-6
-8
- 10
MT  (4  (6))2  (4  2)2
HA  (6  4)2  (4  2)2
MT  136
HA  136
2.
 8  6 6  8 
Midpo int AB  
,
   (1, 7 
2 
 2
Dis tan ce 
 0  7    0  (1) 
2
 50  5 2
2
3.
y
10
8
6
4
D(2, -9)
2
- 10
-8
-6
-4
-2
-2
A(8, 0)
2
4
6
8
10
x
-4
C(-1, -7)
-6
-8
- 10
Length  117
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units;
B(5, 2)
Width  13 units
287
Area  1521 square units
2007-2008
Mathematics 3204/05
Public Exam Questions
4. PQ  17
QR  29
PR  72
Formulae/Proofs
5. Midpo int of M  (3, 4)
AM  BM  CM  5 units
6. mTQ 
2  (4)
6

 3
4  (2) 2
Midpo int of TQ   3, 1
Slope of segment joining midpoint of TQ and origin =
0  (1) 1

0  (3) 3
Slopes are negative reciprocals of each other therefore segments are perpendicular
7. mBD  7
mAC  
1
7
Negative reciprocals segments are perpendicular
AD  50 units  CD
BD is the perpendicular bisector of AC
 2  (4) 3  5 
 
,
   1, 1
2
2 

8. Coordinates of M
 4  8 5  3 
Coordinates of N  
,
   2, 4 
2 
 2
MN  18  3 2
AB  72  6 2
MN  12 AB
9. AC  BD  50
10. AB  5 units
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BC  5units
AC  50 units
288
2007-2008
Mathematics 3204/05
Public Exam Questions
Formulae/Proofs
Mathematics
3205
Section
Labrador School Board
289
2007-2008
Mathematics 3204/05
Public Exam Questions
Formulae/Proofs
1. What is the distance between R and the midpoint of PQ ?
y
(A)
(B)
(C)
(D)
2a 2
a 2
a 10
10a 2
R (2a, 4a)
S
Q (2a, 2a)
P
x
2.
On a coordinate plane, S  0, 5a  and T  2a, 0 are the endpoints of a diameter of a
circle. What is the radius of the circle?
(A)
26
2
(B)
29
2
26
2
29
2
(C)
(D)
3.
a
a
a
a
Given P  a, b  and Q  3a, 5b  , what is the distance from R  3a, 2b  to the
midpoint of PQ ?
(A)
a b
(B)
(C)
a 2  b2
5a  5b
a 2  16b2
(D)
4.
Given the endpoints of a diameter of a circle are A  2 x, 5 y  and B  6 x, y  , what
is the radius of the circle?
(A)
x2  y2
(B)
2 x2  y2
(C)
4 x2  y2
(D)
8 x2  y 2
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Mathematics 3204/05
5.
Public Exam Questions
Formulae/Proofs
What is the length of chord BD in the circle with centre O shown?
B
(A)
(B)
(C)
(D)
6.
2 2
4 2
8
16
D (9,10)
O
A (1,2)
What is the slope of the median from A to BC in the diagram shown?
A(2a,4b)
(A)
(B)
(C)
(D)
2
5ab
2a
C(6a,2b)
5b
5ab
2
5b
2a
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B( - 4a,b)
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Public Exam Questions
Formulae/Proofs
Answers
1. C
2. B
3. B
4. C
5. C
6. D
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Public Exam Questions
Formulae/Proofs
1. Given  MNO as illustrated, prove median MP is equal in length to median NQ .
N (2a, 2a 3 )
P
M (4a,0)
O (0,0)
Q
2. Using the diagram shown, prove that the diagonals of a rectangle are equal in length
and bisect each other.
y
Q (0,b)
R (2a,b)
P (0,0)
S (2a,0)
x
3. Using coordinate geometry, prove that the midpoint of the hypotenuse of a right
triangle is equidistant from all vertices.
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Public Exam Questions
Formulae/Proofs
2– centre
4
6
8
10
2
4
6
4. Determine the equation of the line joining the
of circle P and circle Q if the
2– 6
4
6
8
2
4
circles intersect at the points X  5, 3 and Y  3,1 as shown.
y
10
8
6
4
P
X (5,3)
2
Y (-3,1)
Q
– 6 – 4 – 2
– 2
2
4
6
– 4
– 6
5. Using coordinate geometry, prove that the midpoint of AC is equidistant from all
three vertices.
A (-a, 4a 2)
M
B (-a, 0)
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C (5a, 0)
2007-2008
8 x
Mathematics 3204/05
Public Exam Questions
Formulae/Proofs
6. Using coordinate geometry, prove that the segments joining the midpoints of adjacent
sides of WXYZ form a parallelogram.
y
X (2b,2c)
W (0,0)
Y (2a+2b,2c)
Z (2a,0)
x
7. Use coordinate geometry to prove the segments joining the midpoints of the opposite
sides of any rectangle are perpendicular.
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Public Exam Questions
Formulae/Proofs
Answers
1. MQ  NQ  12a 2  2a 3
2. PR  QS  4a 2  b 2

Midpoint of PR and QS is  a,

3.
b

2
y
(0, b)
(0, 0)
x
(a, 0)
Midpoint of hypotenuse =
 a2 ,
Distance from midpoint to vertices =
b
2

a 2  b2 1 2

a  b2
4
2
4. y  4 x  6

5. Midpoint of AC = 2a, 2a 2
Distance to vertices =
6.

17a 2  a 17
 b, c 
B(midpo int of XY ) is  a  2b, 2c 
C (midpo int of YZ ) is  2a  b, c 
D  midpo int of WZ  is  a, 0) 
A(midpo int of WX ) is
mAB  mDC 
c
ab
mAD  mBC 
c
ba
AB  DC  c 2   a  b   a 2  b 2  c 2  2ab
2
AD  BC  (a  b) 2  c 2  a 2  b 2  c 2  2ab
Labrador School Board
296
2007-2008
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Public Exam Questions
7.
Formulae/Proofs
y
D(a, b)
C(0, b)
A(0, 0)
AC  BD  b units
B(a, 0)
x
AB  CD  a units
mAC  mBD  undefined
mAB  mCD  0
 AC  CD; AC  AB; BD  AB; BD  CD
Labrador School Board
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2007-2008