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Click "Step 1: Sketch" below to begin applying the SOS framework for circumcentre.
Step 1
Sketch a Rough Diagram
First, we plot the given vertices roughly on the coordinate plane to visualize the triangle.
- Point A(4, -2) is in Quadrant IV.
- Point B(4, 10) is in Quadrant I.
- Point C(-6, 8) is in Quadrant II.
Step 2
Observe if there is any vertical or horizontal side
Notice that side AB is a vertical segment because both points A and B have the same x-coordinate (4).
Since the circumcentre lies on the perpendicular bisector of AB, let the circumcentre be K(x, 4).
Denote the mid-point of AC by M1,
M1 =
(
4 + (-6)
2
,
-2 + 8
2
)
=
(-1, 3).
Step 3
Use Slopes to find the remaining coordinate
Mid-point of AC, M1 = (-1, 3)
Slope of AC =
8 - (-2)
-6 - 4
= -1
Slope of KM1 =
4 - 3
x - (-1)
=
1
x + 1
Since KM1 ⊥ AC,
1
x + 1
×
(-1) = -1
x + 1 = 1 ⇒ x = 0
So, the x-coordinate of the circumcentre is 0 (Circumcentre K = (0, 4)).
Summary
-
SSketch a Rough Diagram
-
OObserve horizontal or vertical sides
-
SUse Slopes to find coordinates
Diagram will appear when you begin