A triangle has corners at #(2 , 2 )#, ( 5, 6 )#, and #( 1, 4 )#. What are the endpoints and lengths of the triangle's perpendicular bisectors?
Each of the 3 segments have one endpoint at the midpoint of the given points and the other endpoint at
Thus, line 1 intersects with BC, line 2 intersects with BC, and line 3 intersects with AB. Line 1 is perpendicular to AB.
We require the equations of the three perpendicular lines as well as the lines in which the sides AB and BC lay.
identifying the interceptions on AB and BC sides
Putting equations [b] and [1] together
Putting equations [a] and [2] together
Putting the equations [a] and [3] together
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To find the perpendicular bisectors of the triangle, we need to find the midpoint of each side and then determine the slope of each side. After that, we can find the perpendicular bisectors, which will have slopes negative reciprocals of the slopes of the original sides. Finally, we find the equations of the lines passing through the midpoints with these slopes.

Midpoint of side 1 (between points (2, 2) and (5, 6)): Midpoint ( = \left(\frac{2 + 5}{2}, \frac{2 + 6}{2}\right) = (3.5, 4) )

Midpoint of side 2 (between points (2, 2) and (1, 4)): Midpoint ( = \left(\frac{2 + 1}{2}, \frac{2 + 4}{2}\right) = (1.5, 3) )

Midpoint of side 3 (between points (5, 6) and (1, 4)): Midpoint ( = \left(\frac{5 + 1}{2}, \frac{6 + 4}{2}\right) = (3, 5) )
Now, let's find the slopes of the sides:
 Slope of side 1: ( m_1 = \frac{6  2}{5  2} = \frac{4}{3} )
 Slope of side 2: ( m_2 = \frac{4  2}{1  2} = 2 )
 Slope of side 3: ( m_3 = \frac{4  6}{1  5} = \frac{2}{4} = \frac{1}{2} )
Now, the slopes of the perpendicular bisectors will be negative reciprocals of these slopes:
 Perpendicular bisector 1: Slope ( = \frac{1}{\frac{4}{3}} = \frac{3}{4} )
 Perpendicular bisector 2: Slope ( = \frac{1}{2} = \frac{1}{2} )
 Perpendicular bisector 3: Slope ( = \frac{1}{\frac{1}{2}} = 2 )
Finally, we use the pointslope form to find the equations of the perpendicular bisectors:

Perpendicular bisector 1: ( y  4 = \frac{3}{4}(x  3.5) )

Perpendicular bisector 2: ( y  3 = \frac{1}{2}(x  1.5) )

Perpendicular bisector 3: ( y  5 = 2(x  3) )
These are the equations of the perpendicular bisectors. You can solve them to find their endpoints and lengths.
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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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