Analysing Triangle Vertices and Bisectors

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Part 1

The diagram shows a triangle with vertices O(0,0), A(2,6) and B(12,6). The perpendicular bisectors of OA and AB meet a C.

(a) In order to write down the perpendicular bisector of the line joining the points A(2,6) and B(12,6), I need to find the line's mid-point.

The mid-point of the line joining P(x1, y1) to Q(x2, y2) has the co-ordinates ( )

So the co-ordinates of the midpoint of AB are ( ) = (7,6)

As the two points A(2,6) and B(12,6) have the same y-value, the gradient of the line joining the points is 0. This means that the line's perpendicular bisector also has a gradient of 0.

Thus the equation of the bisector is x = 7

(b) To find the equation of the perpendicular bisector of the line joining the points O (0,0) and A(2,6), I again need to find the co-ordinates of the mid-point of OA. The gradient, and hence that of the perpendicular bisector, can also be found. Thus, knowing the gradient of the perpendicular bisector and one point on it, I can use y - y1 = m(x - x1) (where m is the gradient) to obtain the required equation.

Co-ordinates of the mid-point of OA are ( ) = (1,3)

The gradient of the line joining the points O (0,0) and A(2,6) is a measure of the steepness of the line OA and it is the ratio of the change in the y co-ordinate to the change in the x co-ordinate

in going from O to A.

Thus, the gradient of OA = = = 3

If two lines are perpendicular, the product of their gradients is -1. This condition for perpendicular lines means that is one line has a gradient of m, a perpendicular line will have gradient

It follows that if the gradient of OA = 3, then the gradient of the perpendicular bisector of OA =

The perpendicular bisector has gradient and passes through the point (1,3).

So using y - y1 = m(x - x1) (where m is the gradient), I can find the equation of the line:

The equation of the perpendicular bisector of OA is 3y = 10 - x

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(c) The perpendicular bisectors of the lines OA and AB intersect at C.

Since the point of intersection has co-ordinates that satisfy both equations, it is possible to substitute one of the original equations into the other to show the co-ordinates of C.

Perpendicular bisector of AB is x = 7

Perpendicular bisector of OA is 3y = 10 - x

Substituting x = 7 into 3y = 10 - x gives 3y = 10 - 7

or 3y = 3

or y = 1

Hence showing that the co-ordinates of ...

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