Tak Sun Secondary School Form 3 Mathematics

Worksheet 12.5 Chapter 12 Coordinate Geometry of Straight Lines

Name : ___________________________ Class : _________ No. :_______

1. In the figure, OAC is an isosceles triangle with base 10 units and height 8 units. B is the mid-point

of OA. Write down the coordinates of A, B and C.

∵ OA 10 units ∴ Coordinates of )0 ,10(A

∵ B is the mid-point of OA.

∴ Coordinates of )0 ,5(0 ,2

10

B

∵ OAC is an isosceles triangle with OC AC. ∴ Coordinates of )8 ,5()80 ,5( C

2. Given that the vertices of a parallelogram are A(–2, 4), B(–4, 1), C(–1, –3) and D( 1, 0).

(a) Find the lengths of AB, BC, CD and DA.

(b) Hence, show that the opposite sides of parallelogram ABCD are equal.

(a) 13)41(2)](4[ 22 AB

5)13(4)](1[ 22 BC

133)](0[1)](1[ 22 CD

5)40(2)](1[ 22 DA

(b) ∵ AB CD and BC DA ∴ The opposite sides of parallelogram ABCD are equal.

3. It is given that O(0, 0), A(p, 0), B(p + q, r) and C(q, r) are the vertices of a parallelogram OABC.

Prove that the diagonals OB and AC bisect each other.

Mid-point of

2 ,

2

2

0 ,

2

)(0

rqp

rqpOB

Mid-point of

2 ,

2

2

0 ,

2

rqp

rqpAC

∵ Mid-point of OB and of AC is the same point. ∴ The diagonals OB and AC bisect each other.

4. In the figure, P(0, p), Q and R(r, 0) are the vertices of △PQR. Given that OP is the perpendicular

bisector of QR, prove that △PQR is an isosceles triangle.

∵ OP is the perpendicular bisector of QR. ∴ OP QR and QO OR ∴ Coordinates of Q (–r, 0)

units

units )0()0(

units

units )0()0(

22

22

22

22

pr

prPR

pr

prPQ

∵ PQ PR ∴ △PQR is an isosceles triangle.

5. In the figure, OA is the base of rhombus OACB. It is given that the height of rhombus OACB is 4

units.

(a) Find the coordinates of B and C.

(b) Hence, prove that AB⊥CO.

(a) ∵ Coordinates of A = (5, 0) ∴ The side of the rhombus is 5 units. With the notations in the figure, consider △ODB where ∠ODB = 90.

units3

units45

Theorem) Pyth.(

22

222

OD

BODBOD

∵ OACB is a rhombus. ∴ Coordinates of B = (3, 4), and Coordinates of C = (3+5, 4) = (8, 4)

(b) Slope of2

1

08

04

CO

Slope of 253

04

AB

∵ Slope of AB slope of CO 2 ×2

1= 1

∴ AB CO

6. By the analytic approach, prove that if the diagonals of a rectangle are perpendicular to each other,

then the rectangle is a square.

Let a and b be the length and the width of a rectangle respectively. Set up the coordinate system as shown in the figure. The vertices of the rectangle

are A(a, 0), B(a, b), C(0, b) and D(0, 0).

a

b

a

bDB

0

0 of Slope

a

b

a

bAC

0

0 of Slope

∵ OB AC

∴

(rejected) or

1

22

aab

ab

a

b

a

b

∴ The rectangle is a square.

7. By the analytic approach, prove that if a line passes through the mid-point of one side of a

triangle and is parallel to another side, it bisects the third side of the triangle.

Let O(0, 0), P(a, b) and Q(c, 0) be the vertices of the triangle. Set up the coordinate system as shown in the figure. M is the mid-point of OP. N(x, y) is a point on PQ such that MN // OQ.

2

,2

of sCoordinateba

M

∵ MN // OQ

∴ y-coordinate of N = y-coordinate of M =2

b

∵ P, N and Q are collinear. ∴ Slope of PQ slope of NQ

2

)(2

20

0

cax

acxcxc

b

ac

b

∴ Coordinates of

2

,2

bcaN , which is the mid-point of PQ.

∴ If the line segment passing through the mid-point of one side of a triangle and parallel to another side, it bisects the third side of the triangle.

8. By the analytic approach, prove that the line segments joining the mid-points of opposite sides of

any quadrilateral bisect each other.

Coordinates of P )0 ,(0 ,2

02a

a

Coordinates of Q ) ,(2

20 ,

2

22cba

cba

Coordinates of R ) ,(2

22 ,

2

22ecdb

ecdb

Coordinates of S ) ,(2

02 ,

2

02ed

ed

Coordinates of the mid-point of PR

2 ,

2

2

)(0 ,

2

)(

ecdba

ecdba

Coordinates of the mid-point of QS

2 ,

2

2 ,

2

)(

ecdba

ecdba

∴ The mid-points of PR and QS coincide. ∴ PR and QS bisect each other. ∴ The line segments joining the mid-points of opposite sides of any quadrilateral bisect each other.