Question

A(1,7),B(7,5) and C(0,-2),find the point of intersection of BC with the perpendicular bisector of AB ?

Answer 1

`(2,0)`

Explanation:

This will be pretty long. I'm not sure if there's a shorter solution, but this is how I would solve it.

We will start by finding the equation of the perpendicular bisector of `bar(AB)`.

Step 1 - Slope of the perpendicular bisector.

We will first find the slope of `bar(AB)` using the formula for slope.
`(y_1-y_2)/(x_1-x_2)`
`=[(7)-(5)]/[(1)-(7)]`
`=(2)/(-6)`
`=-1/3`

We will now look for the slope of the perpendicular bisector. To do this, we will get the negative reciprocal of the slope of `bar(AB)`
Slope of `bar(AB)`: `(-1/3)`
Negative reciprocal: `3`

The slope of the perpendicular bisector is 3.

Step 2 - Point of intersection of `bar(AB)` and the perpendicular bisector.

Since this is a bisector we're talking about, it should go through the middle of `bar(AB)`. To solve for the point of intersection, we will use the midpoint formula.
`((x_1+x_2)/2,(y_1+y_2)/2)`
`((1+7)/2,(7+5)/2)`
`(8/2,12/2)`
`(4,6)`

The point of intersection between `bar(AB)` and the perpendicular bisector is (4,6).

Step 3 - The equation of the perpendicular bisector.

Since we know the slope of the bisector and one point it intersects, we can write the equation of the line in point-slope form.
`y-y_0=m(x-x_0)`
`y-(6)=(3)[x-(4)]`
`y-6=3x-12`
`y=3x-12+6`
`y=3x-6`

The equation of the perpendicular bisector is `y=3x-6`.

We are done with the perpendicular bisector, so we can now proceed to finding the equation of the line that includes `bar(BC)`.

Step 1 - Slope of `bar(BC)`.
`(y_1-y_2)/(x_1-x_2)`
`=[(5)-(-2)]/[(7)-(0)]`
`=(7)/(7)`
`=1`

The slope of `bar(BC)` is 1.

Step 2 - Equation of the line including `bar(BC)`.

We know that the line passes through (7,5) and (0,-2), so we can write the equation in point-slope form using either of the two points. I will use (0,-2) since it is simpler.
`y-y_0=m(x-x_0)`
`y-(-2)=(1)[x-(0)]`
`y+2=x`
`y=x-2`

The equation of the line including `bar(BC)` is `y=x-2`.

FINAL STEP
Now that we have both the equations of the perpendicular bisector and the line including `bar(BC)`, we can now look for the point of intersection using substitution.
`y=x-2`
`3x-6=x-2`
`3x-x=-2+6`
`2x=4`
`x=2`

To solve for the value of `y`, just substitute the value `2` to either of the two equations.
`y=x-2`
`y=(2)-2`
`y=0`

The point of intersection of `bar(BC)` and the perpendicular bisector is `(2,0)`.