A triangle has corners at #(4 , 1 )#, ( 2, 4 )#, and #( 0, 2 )#. What are the endpoints of the triangle's perpendicular bisectors?
The easy endpoints are the midpoints,
By the perpendicular bisectors of a triangle we presumably mean the perpendicular bisector of each side of a triangle. So there are three perpendicular bisectors for every triangle.
Each perpendicular bisector is defined to intersect one side at its midpoint. It will also intersect one of the other sides. We'll presume those two meets are the endpoints.
The midpoints are
This is probably a good place to learn about parametric representations for lines and line segments.
Let's label the points
As
Let's work one out.
The direction vector from C to B is
(Different line, different parameter.) We can see where this meets each of the sides.
Subtracting,
That's outside the range so the perpendicular bisector of BC doesn't hit the side AB.
Subtracting,
That gives the other endpoint as
This is getting long, so I'll leave the other two endpoints to you.
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To find the endpoints of the perpendicular bisectors of the triangle, we first need to determine the midpoints of each side of the triangle, and then find the slopes of the lines perpendicular to each side passing through these midpoints.

For the side connecting points (4, 1) and (2, 4), the midpoint is ((4+2)/2, (1+4)/2) = (3, 2.5). The slope of this side is (4  1) / (2  4) = 3 / 2 = 1.5. Therefore, the slope of the perpendicular bisector will be the negative reciprocal of 1.5, which is 2/3. Using the midpoint (3, 2.5) and slope 2/3, we can find the equation of the perpendicular bisector.

For the side connecting points (2, 4) and (0, 2), the midpoint is ((2+0)/2, (4+2)/2) = (1, 3). The slope of this side is (4  2) / (2  0) = 2 / 2 = 1. Therefore, the slope of the perpendicular bisector will be the negative reciprocal of 1, which is 1. Using the midpoint (1, 3) and slope 1, we can find the equation of the perpendicular bisector.

For the side connecting points (0, 2) and (4, 1), the midpoint is ((0+4)/2, (2+1)/2) = (2, 1.5). The slope of this side is (1  2) / (4  0) = 1 / 4. Therefore, the slope of the perpendicular bisector will be the negative reciprocal of 1/4, which is 4. Using the midpoint (2, 1.5) and slope 4, we can find the equation of the perpendicular bisector.
Once we have the equations of the perpendicular bisectors, we can find their intersections to determine the endpoints. These intersections will be the endpoints of the perpendicular bisectors of the triangle.
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The endpoints of the triangle's perpendicular bisectors are:
 Perpendicular bisector of the segment joining (4, 1) and (2, 4): (3, 2.5) and (1, 2.5)
 Perpendicular bisector of the segment joining (2, 4) and (0, 2): (1, 3) and (1, 1)
 Perpendicular bisector of the segment joining (0, 2) and (4, 1): (2, 1.5) and (2, 2.5)
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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