Triangle ABC is shown in the diagram below. The coordinates of B are (3,0) and the coordinates of C are (9,-2). The broken line is the perpendicular bisector of BC. (a) Find the equation of the perpendicular bisector of BC. (b) The line AB makes an angle of 45° with the positive direction of the x-axis. Find the equation of AB. (c) Find the coordinates of the point of intersection of AB and the perpendicular bisector of BC.

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Triangle ABC is shown in the diagram below - Scottish Highers Maths - Question 1 - 2017

Question 1

Triangle ABC is shown in the diagram below. The coordinates of B are (3,0) and the coordinates of C are (9,-2). The broken line is the perpendicular bisector of ... show full transcript

Worked Solution & Example Answer:Triangle ABC is shown in the diagram below - Scottish Highers Maths - Question 1 - 2017

Step 1

Find the equation of the perpendicular bisector of BC.

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Answer

First, calculate the midpoint of segment BC.
The coordinates of B are (3,0) and C are (9,-2).
The formula for the midpoint, M, between two points (x1, y1) and (x2, y2) is:

$M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) = \left( \frac{3 + 9}{2}, \frac{0 - 2}{2} \right) = (6, -1)$
Next, calculate the gradient (slope) of line BC.
The slope, m, between points (x1, y1) and (x2, y2) is given by:

$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-2 - 0}{9 - 3} = \frac{-2}{6} = -\frac{1}{3}$
The gradient of the perpendicular bisector is the negative reciprocal:

$m_{perpendicular} = -\frac{1}{m} = -\frac{1}{-\frac{1}{3}} = 3$
Now, using the point-slope form of the line equation (y - y_1 = m(x - x_1)), with the midpoint (6, -1) and the slope 3, the equation becomes:

$y - (-1) = 3(x - 6)$
Simplifying gives:
$y + 1 = 3x - 18 \Rightarrow y = 3x - 19$

Thus, the equation of the perpendicular bisector is:
$y = 3x - 19$

Step 2

Find the equation of AB.

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Answer

The line AB makes an angle of 45° with the positive x-axis, thus the slope m is given by:

$m = \tan(45°) = 1$
Using the point-slope form with point A (9, -2):

$y - (-2) = 1(x - 3)$
Which simplifies to:
$y + 2 = x - 3 \Rightarrow y = x - 5$

Hence, the equation of AB is:
$y = x - 5$

Step 3

Find the coordinates of the point of intersection of AB and the perpendicular bisector of BC.

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Answer

To find the intersection, set the equations equal:

$3x - 19 = x - 5$
Rearranging gives:

$3x - x = 19 - 5 \Rightarrow 2x = 14 \Rightarrow x = 7$
Substitute x back into either equation to find y:
Using AB's equation:
$y = 7 - 5 = 2$

The coordinates of the point of intersection are:
((7, 2))

Triangle ABC is shown in the diagram below - Scottish Highers Maths - Question 1 - 2017

Worked Solution & Example Answer:Triangle ABC is shown in the diagram below - Scottish Highers Maths - Question 1 - 2017

Find the equation of the perpendicular bisector of BC.

Find the equation of AB.

Find the coordinates of the point of intersection of AB and the perpendicular bisector of BC.

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