Everything Maths Grade 10

31.Given the following diagram:

 2 2 4 6 8

 2

2

4

6

A(2; 2)

B(4; 5)

C(8; 2)

D(1;3)

E(a;b)

x

y

a)IfEis the mid-point ofAB, find the values ofaandb.

b)Find the equation of the line perpendicular toBC, which passes through the origin.

c)Find the coordinates of the mid-point of diagonalBD.

d)Hence show thatABCDis not a parallelogram.

e)IfCcould be moved, give its new coordinates so thatABCDwould be a parallelogram.

32.A triangle has verticesA(1; 7),B(8; 4)andC(5;5).

a)Calculate the gradient ofAB.

b)Prove that the triangle is right-angled atB.

c)Determine the length ofAB.

d)Determine the equation of the line fromAto the mid-point ofBC.

e)Find the area of the triangleABC.

33.A quadrilateral has verticesA(0; 5),B(3;4),C(0;5)andD(4;k)wherek 0.

a)What shouldkbe so thatADis parallel toCD?

b)What shouldkbe so thatCD=

p

52?

34.On the Cartesian plane, the three pointsP(3; 4),Q(7;1)andR(3;b)are collinear.

a)Find the length ofP Q.

b)Find the gradient ofP Q.

c)Find the equation ofP Q.

d)Find the value ofb.

35.GivenA(4; 9)andB(2;3).

a)Find the mid-pointMofAB.

b)Find the gradient ofAB.

c)Find the gradient of the line perpendicular toAB.

d)Find the equation of the perpendicular bisector ofAB.

e)Find the equation of the line parallel toAB, passing through(0; 6).

36.L(1;1),M(2; 4),N(x;y)andP(4; 0)are the vertices of parallelogramLM N P.

a)Determine the coordinates ofN.

b)Show thatM Pis perpendicular toLNand state what type of quadrilateralLM N Pis, other than a

parallelogram.

c)Show thatLM N Pis a square.

Chapter 8. Analytical geometry 323