Linear functions

[d0minicz]

Three points have the coodinates A(1,7), B(7,5) and C (0,-2). Find:
a) the equation of the perpendicular bisector of AB
b) the point of intersection of this perpendicular bisector and BC.
Thanks

[Flaming_Arrow]

a) to find that, find the midpoint of AB then find the gradient of AB then use the perpendicular rule to find the equation.

b) Perpendicular Bisector equation = equation of BC

[kj_]

a) To find the perpendicular bisector of AB, we first have to determine the equation of the line AB.
A(1,7) and B(7,5), and thus finding the gradient of ,


We also need a point - the midpoint of line AB.
The x-coordinate is defined by

The y-coordinate is defined by
Thus the midpoint is M(4,6)

Now, the perpendicular bisector passes through the midpoint and its' gradient is normal to the original line AB.
Thus,

Substituting this and the midpoint M(4,6) into the equation ,



Thus the equation of the perpendicular bisector of AB is

b) B(7,5) and C(0,-2), we need the equation of the line BC and thus we need to work out the gradient and y-intercept of BC.



We are given C(0,-2) so we know that the y-intercept and the value of c in is .

Thus,

Equating these two equations,






Thus, substituting into either equation, we obtain

Therefore, the point of intersection of the perpendicular bisector and BC is

[d0minicz]

The point (h,k) lies on the line y=x+1 and is 5 units from the point (0,2). Write down two equations connecting h and k and hence find the possible values of h and k. Thanks

[Flaming_Arrow]

and

i think you can work it out from there

[d0minicz]

P and Q are the points of intersection of the line y/2 + x/3 = 1 with the x and y axes respectively. The gradient of QR is 1/2, where R is the point with the x-coordinate 2a, where a>0.
a) find the y-coordinate of R in terms of a
b) find the value of a if the gradient of PR is -2.
Much appreciated

[Flaming_Arrow]

a.

gradient of QR  where where

and

use gradient formula to get the y coordinate of R







b. gradient of PR = -2





then we have our two equations to find the value of a






this question is worded very trickily, make sure you pick up how it says x and y respectively to know the values of P and Q

[d0minicz]

Triangle ABC has the coordinates A(1,1) and B(-1,4). The gradients of AB, AC and BC are -3m, 3m and m respectively.
a) Find the coordinates of C.
b) SHow that AC=2AB

Thankz

[Flaming_Arrow]

Gradient of AB = -3m(in terms of m)









to find C, we know what ABC is a triangle, meaning AC and BC intersect at point C

Gradient of AC =

Gradient of BC =


Equation of AC





Equation of BC






make then equal each other to find the intersection which is C







sub x into any of the 2 equations to get Y








b.

find the length of AC



find the length of AB

[d0minicz]

In the rectangle ABCD, A and B are the points (4,2) and (2,8) respectively. Given that the equation of AC is y=x-2, find:
a) the equation of BC
b) the coordinates of C
c) the coordinates of D
d) the area of rectangle ABCD

Alright, I'm having trouble finding unknown coordinates and such, so can someone please guide me through on how to approach these questions and types of methods that can be used to find unknown gradients/coordinates etc. Thanks alot

[Flaming_Arrow]

hint: to find the equation of BC find the gradient of AB then use the perpendicular rule then sub in point B to get the equation of BC

[d0minicz]

Thanks but why do you use the perpendicular rule, and could you give me hints to find the coordinates of C etc. Thank you

[Flaming_Arrow]

because AB is perpendicular to BC, AC and BC will meet at C so make  then equal each other to find coordinates of C

[d0minicz]

ABCD is a parallelogram, lettered anticlockwise, such that A and C are the points (-1,5) and (5,1) respectively.
b) Given that BD is parallel to the line whose equation is y + 5x = 2, find the equation of BD.
c) Given that BD is perpendicular to AC, find:
i) the equation of BC
ii) the coordinates of B
iii) the coordinates of D

Any hints please, appreciate it

edit: sorry for posting the question up, not thinking straight, thanks for reply anyway

[Flaming_Arrow]

ii) BC and BD intersect at B