Medians and Perpendicular bisectors:

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2.10 Using Point of Intersection to Solve Problems
Medians and Perpendicular bisectors
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Ex. 1 The coordinates of DABC are A(4, 7), B(2,
3), C(6, 1). Find the intersection of the
medians (centroid).
Find the equation of the median from B to AC.
Find the midpoint of AC.
MAC (5, 3)
Find slope of BD
mBD 0
3
mBD 0
y mx b
y 0x b
y b
y 3
(equation of BD)
Part 2
Find the equation ofthe median from A.
Find the midpoint of BC.
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Find the equation of the median from A.
Find the midpoint of BC.
MBC (2, 1)
Find the slope of AE.
mAE 3
y 3x b
1 3(2) b
5 b
5
y 3x 5
equation of AE
equation of BD
y 3
solve by substitution
3x 5 3
3x 3 5
3x 8
x 2.67
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The centroid is the center of gravity of the
triangle.
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Ex. 2 The coordinates of A(0, 5), B(8, 3), and
C(6, 5). Find the circumcentre. (intersection
of the perpendicular bisectors of the sides.
Find the perpendicular bisector of BC.
Find the midpoint of BC.
C(6, 5)
B(8, 3)
MBC (7, 4)
Find the slope of BC.
1
A(0, 5)
8
mBC 1
Slope of perpendicular bisector is 1
MBC (7, 4)
y mx b
4 (1)7 b
4 7 b
C(6, 5)
3 b
B(8, 3)
y x 3
(perpendicular bisector of BC)
A(0, 5)
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Find the perpendicular bisector of AB
Find the midpoint of AB
MAB (4, 1)
Find the slope of AB.
C(6, 5)
B(8, 3)
mAB 1
The slope of the perpendicular bisector is 1
A(0, 5)
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The slope of the perpendicular bisector is 1
MAB (4, 1)
y mx b
1 (1) 4 b
1 4 b
C(6, 5)
3 b
B(8, 3)
y x 3
Perpendicular bisector of AB.
A(0, 5)
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Perpendicular bisector of AB.
(i) y x 3
(ii) y x 3
Perpendicular bisector of BC.
2y 0
y 0
C(6, 5)
sub y 0 into (i)
B(8, 3)
x 3
A(0, 5)
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The circumcentre is the same distance from the
three vertices of the triangle
C(6, 5)
B(8, 3)
(3, 0)
A(0, 5)