A triangle has corners at #(2 , 5 )#, ( 1, 3 )#, and #( 8, 1 )#. What are the endpoints and lengths of the triangle's perpendicular bisectors?
I didn't finish, but here's some interesting stuff.
Given that it is two years old, why does this appear on "just asked"?
I will attempt to improve upon my recent lengthy, meandering partial response to one similar to this one.
Since I'm not familiar with matrices, let's just focus on the mapping.
Just so you know, that's a cross product and a dot product. Now let's examine our triangle under R:
The question has an obvious answer because in the transformed space, the y axis is the perpendicular bisector of AB. Depending on the sign of ac+bd, the bisector hits either B'C' (positive) or A'C' (negative).
The opposite conversion to R is
Now let's map our intercept back to y:
That's pretty awesome. Some problems are too big for a short answer, so I'm just going to post this without finishing because I'm getting warnings that the answer is too long.
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The endpoints of the perpendicular bisectors are as follows:

For the line passing through the midpoint of the segment between (2, 5) and (1, 3): Endpoint 1: (3.5, 4) Endpoint 2: (4, 4)

For the line passing through the midpoint of the segment between (2, 5) and (8, 1): Endpoint 1: (5, 3) Endpoint 2: (5, 3)

For the line passing through the midpoint of the segment between (1, 3) and (8, 1): Endpoint 1: (4.5, 2) Endpoint 2: (4.5, 2)
The lengths of the perpendicular bisectors can be calculated using the distance formula between the endpoints.
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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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