1. Grade 10 Academic (MPM2D) Unit 2: Analytic Geometry Medians and Centroid
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
Medians and Centroid
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3. Median & Centroid Medians and Centroid Medians:
line from vertex to the midpoint of the opposite side
Centroid:
where all 3 medians meet
divides each median in the ratio 1: 2
it is the centre of mass
Medians and Centroid
© 2017 E. Choi – MPM2D - All Rights Reserved

4. Instructions Medians and Centroid Plot the following points on the
Cartesian plane given below.
A(-1, 4) B(7, 2) C(1, -6)
b) Determine the point D, the mid
point of BC.
c) Draw the line through A and D.
What is this line called?
Determine the equation of this line. (Use the space below)
Repeat steps b to e for the other 2 medians BE & CF.
Prove that the medians of a triangle pass through the same point.
h) What is that point called?
Medians and Centroid
© 2017 E. Choi – MPM2D - All Rights Reserved

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

6. Continued b) Determine the point D, the mid point of BC.
c) Draw the line through A and D.
What is this line called?
Determine the equation of this line. (Use the space below)
Median
Midpoint of BC  D
Equation of Median AD
D (4, -2)
Medians and Centroid
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7. Continued f) Repeat steps b to e for the other 2 medians BE & CF.
Midpoint of AC  E
Equation of Median BE
E (0, -1)
D (4, -2)
Medians and Centroid
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8. Continued f) Repeat steps b to e for the other 2 medians BE & CF.
Midpoint of AB  F
F (3, 3)
Equation of Median CF
E (0, -1)
D (4, -2)
Medians and Centroid
© 2017 E. Choi – MPM2D - All Rights Reserved

9. Continued: Point of Intersection (Centroid)
g) Prove that the medians 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 medians intersect at the same point.
Recall:
Let (1) = (2)
F (3, 3)
E (0, -1)
D (4, -2)
Medians and Centroid
© 2017 E. Choi – MPM2D - All Rights Reserved

10. Continued: Point of Intersection (Centroid)
g) Prove that the medians of a triangle pass through the same point.
Check by substitute
Recall:
F (3, 3)
E (0, -1)
Therefore all 3 median intersect at the same point
 Centroid:
D (4, -2)
Medians and Centroid
© 2017 E. Choi – MPM2D - All Rights Reserved

11. Centroid Formula!! Let’s redo the same example by introducing the Centroid formula
Recall:
A(-1, 4) B(7, 2) C(1, -6)
F (3, 3)
E (0, -1)
D (4, -2)
Same!!!
Medians and Centroid
© 2017 E. Choi – MPM2D - All Rights Reserved

12. Median & Centroid centre of mass balance point. Medians and Centroid
line from vertex to the midpoint of the opposite side
Centroid:
where all 3 medians meet
divides each median in the ratio 1: 2
it is the centre of mass
Centroid:
The point where the three medians of a triangle intersect. Also called the
centre of mass or
balance point.
Medians and Centroid
© 2017 E. Choi – MPM2D - All Rights Reserved

13. Example 2: Centroid and Median ratios
Centroid is where all 3 medians meet, which also divides each median in the ratio 1: 2.
A(0, 6) B(-6, -6) C(6, -6)
By observation, it is obvious that the
midpoint of BC is:
E (0, -6)
(0, -6)  E
Without any calculations, we can see the length of the median AE is: 12 8
The length of AM is: 4
and the length of ME is:
M (0, -2)
which the centroid divides the median in the ratio of
4:8  1:2
You can always use the distance formula to verify them!!
Feel free to repeat the same procedures to check the ratios of the other two medians from B and C.
Medians and Centroid
© 2017 E. Choi – MPM2D - All Rights Reserved

14. Homework Work sheet: Extra practice #1a-d Text: P. 173 #2, #4, 13,14
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Medians and Centroid
© 2017 E. Choi – MPM2D - All Rights Reserved

15. End of lesson Medians and Centroid
© 2017 E. Choi – MPM2D - All Rights Reserved