1. Grade 10 Academic (MPM2D) Unit 2: Analytic Geometry Perpendicular Bisector & Circumcentre
Mr. Choi
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2. Properties of Triangles
Medians:
line from vertex to the midpoint of the opposite side
Midpoint:
point on a line that divides it into 2 equal parts
to draw a median  find mid point of opposite side and connect to vertex
Altitude:
shows the height of a polygon
to draw an altitude  line from vertex to opposite side so that it meets at 90°
Perpendicular bisectors:
perpendicular to a line segment and meets at its midpoint.
to draw a perpendicular bisector  find midpoint, then measure a right angle, draw line.
Angle bisectors:
The (interior) bisector of an angle, also called the internal angle bisector is the line or line segment that divides the angle into two equal parts 
Centroid:
where all 3 medians meet
divides each median in the ratio 1: 2
it is the centre of mass
Orthocentre:
where all 3 altitudes meet
Circumcentre:
where all 3 perpendicular bisectors meet centre of the circle that passes through the vertices of the triangle
the circle is called circumcircle or circumscribed circle
Incentre:
where all 3 angle bisectors meet centre of the circle that meets each side once
the circle is called incircle or inscribed circle
Perpendicular Bisector & Circumcentre
© 2017 E. Choi – MPM2D - All Rights Reserved

3. Perpendicular Bisector & Circumcentre
Perpendicular bisectors:
perpendicular to a line segment and meets at its midpoint.
to draw a perpendicular bisector  find midpoint, then measure a right angle, draw line.
Circumcentre:
where all 3 perpendicular bisectors meet centre of the circle that passes through the vertices of the triangle
the circle is called circumcircle or circumscribed circle
perpendicular bisector
circumcentre
circumcircle
Perpendicular Bisector & Circumcentre
© 2017 E. Choi – MPM2D - All Rights Reserved

4. Instructions Plot the following points on the
Cartesian plane given below.
A(-1, 4) B(7, 2) C(1, -6)
b) Draw a line perpendicular to BC from D.
c) Determine the slope and equation of this line.
d) What is this line called?
e) Repeat steps (b) to (e) for the other two sides.
f) Prove that the Perpendicular bisectors of a triangle pass through the same point.
g) What is that point called?
Perpendicular Bisector & Circumcentre
© 2017 E. Choi – MPM2D - All Rights Reserved

5. Instructions Plot the following points on the
Cartesian plane given below A(-1, 4) B(7, 2) C(1, -6)
Perpendicular Bisector & Circumcentre
© 2017 E. Choi – MPM2D - All Rights Reserved

6. Continued b) Draw a line perpendicular to BC from D.
c) Determine the slope and equation of this line.
d) What is this line called?
Perpendicular bisector
Midpoint of BC  D
Equation of Perpendicular Bisector from D
Substitute D (4,-2)
D (4, -2)
Negative
reciprocal
Perpendicular Bisector & Circumcentre
© 2017 E. Choi – MPM2D - All Rights Reserved

7. Continued Repeat steps b to e for the other 2 perpendicular bisectors
from E and F.
Midpoint of AC  E
Equation of Perpendicular Bisector from E
Substitute E (0,-1)
E (0, -1)
D (4, -2)
Negative
reciprocal
Perpendicular Bisector & Circumcentre
© 2017 E. Choi – MPM2D - All Rights Reserved

8. Continued
f) Repeat steps b to e for the other 2 perpendicular bisectors
from E and F.
Midpoint of AB  F
Equation of Perpendicular Bisector from F
F (3, 3)
Substitute F (3,3)
E (0, -1)
D (4, -2)
Negative
reciprocal
Perpendicular Bisector & Circumcentre
© 2017 E. Choi – MPM2D - All Rights Reserved

9. Continued: Point of Intersection (Circumcentre)
Prove that the perpendicular bisectors of a triangle pass through
the same point.
By method of comparison with equation 1 and 2, find the point of intersection and check with all three equations. Verify that all perpendicular bisectors intersect at the same point.
Recall:
Let (1) = (2)
F (3, 3)
E (0, -1)
D (4, -2)
Perpendicular Bisector & Circumcentre
© 2017 E. Choi – MPM2D - All Rights Reserved

10. Continued: Point of Intersection (Circumcentre)
Prove that the perpendicular bisectors of a triangle pass through the same point.
Check by substitute
Recall:
F (3, 3)
E (0, -1)
Therefore all 3 perpendicular bisectors intersect at the same point
 Circumcentre:
D (4, -2)
Perpendicular Bisector & Circumcentre
© 2017 E. Choi – MPM2D - All Rights Reserved

11. Perpendicular Bisector & Circumcentre
Perpendicular bisectors:
perpendicular to a line segment and meets at its midpoint.
to draw a perpendicular bisector  find midpoint, then measure a right angle, draw line.
Circumcentre:
where all 3 perpendicular bisectors meet centre of the circle that passes through the vertices of the triangle
the circle is called circumcircle or circumscribed circle
perpendicular bisector
circumcentre
circumcircle
Perpendicular Bisector & Circumcentre
© 2017 E. Choi – MPM2D - All Rights Reserved

12. Continued: (Circumcircle)
h) Determine the equation of the circumcircle
Recall:
F (3, 3)
E (0, -1)
D (4, -2)
Perpendicular Bisector & Circumcentre
© 2017 E. Choi – MPM2D - All Rights Reserved

13. Homework Work sheet: Extra practice #1a-d Text:
Check the website for updates
Perpendicular Bisector & Circumcentre
© 2017 E. Choi – MPM2D - All Rights Reserved

14. End of lesson Perpendicular Bisector & Circumcentre
© 2017 E. Choi – MPM2D - All Rights Reserved