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

##### 1 Answer

#### 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

**Step 1** - Slope of the perpendicular bisector.

We will first find the slope of

We will now look for the slope of the perpendicular bisector. To do this, we will get the negative reciprocal of the slope of

Slope of

Negative reciprocal:

**The slope of the perpendicular bisector is 3.**

**Step 2** - Point of intersection of

Since this is a bisector we're talking about, it should go through the middle of

**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.

**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

**Step 1** - Slope of

**The slope of #bar(BC)# is 1.**

**Step 2** - Equation of the line including

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.

**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

To solve for the value of

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