Everything Maths Grade 10

23.EF GHis a parallelogram with verticesE(1; 2),F(2;1)andG(2; 0). Find the coordinates ofHby
using the fact that the diagonals of a parallelogram bisect each other.

24.P QRSis a quadrilateral with pointsP(0;3),Q(2; 5),R(3; 2)andS(3;2)in the Cartesian plane.

a)Find the length ofQR.

b)Find the gradient ofP S.

c)Find the mid-point ofP R.

d)IsP QRSa parallelogram? Give reasons for your answer.

25.Consider triangleABCwith verticesA(1; 3),B(4; 1)andC(6; 4).

a)Sketch triangleABCon the Cartesian plane.

b)Show thatABCis an isosceles triangle.

c)Determine the coordinates ofM, the mid-point ofAC.

d)Determine the gradient ofAB.

e)Show thatD(7;1)lies on the line that goes throughAandB.

26.△P QRhas verticesP(1; 8),Q(8; 7)andR(7; 0). Show through calculation that△P QRis a right angled
isosceles triangle.
27.△ABChas verticesA(3; 4),B(3;2)andC(5;2).Mis the mid-point ofACandNis the mid-point
ofBC. Use△ABCto prove the mid-point theorem using analytical geometry methods.

28. a)List two properties of a parallelogram.

b)The pointsA(2;4),B(4; 1),C(2; 4)andD(4;1)are the vertices of a quadrilateral. Show that

the quadrilateral is a parallelogram.

29.The diagram shows a quadrilateral. The pointsBandDhave the coordinates(2; 6)and(4; 2)respectively.
The diagonals ofABCDbisect each other at right angles.Fis the point of intersection of lineACwith
they-axis.

 6  4  2 2 4 6 8 10 12

2

4

6

8

A(5; 0)

B(2; 6)

C

D(4; 2)

F

x

y

E

a)Determine the gradient ofAC.

b)Show that the equation ofACis given by 2 y=x+ 5.

c)Determine the coordinates ofC.

30.A(4;1),B(6;3)andC(2; 3)are the vertices of△ABC.

a)Find the length ofBC, correct to 1 decimal place.

b)Calculate the gradient ofAC.

c)IfPhas coordinates(26; 19), show thatA,CandPare collinear.

d)Determine the equation of lineBC.

e)Show that△ABCis right-angled.

322 8.5. Chapter summary