1. AS Test

2. Day 6: Centroids and Circumcentres
Unit 3: Coordinate Geometry
Did you know that the only continent without an active volcano is Australia?

3. Learning Goals
To be able to calculate the centroid and circumcentre of a triangle

4. Centroid
The point of intersection of the three medians of a triangle

5. Draw a triangle with vertices A (0, 4), B (−2, 2) and C (6, 2)
Draw a triangle with vertices A (0, 4), B (−2, 2) and C (6, 2). Sketch the medians. Find the coordinates of the centroid. A B C

6. Draw a triangle with vertices A (0, 4), B (−2, 2) and C (6, 2)
Draw a triangle with vertices A (0, 4), B (−2, 2) and C (6, 2). Sketch the medians. Find the coordinates of the centroid. Step 1: Find the equation of one median A B C
𝑀 𝐴𝐵 = −2 2 ,
𝑀 𝐴𝐵 = − 2 2 , 6 2
𝑀 𝐴𝐵 = −1, 3
𝑚 𝐶𝑀 = 3 − 2 −1 − 6
𝑚 𝐶𝑀 = 1 −7
𝑦=− 1 7 𝑥+𝑏
3 =− 1 7 −1 +𝑏
3= 1 7 +𝑏
3− 1 7 =𝑏
𝑏= or 20 7
𝑦=− 1 7 𝑥+ 20 7

7. Draw a triangle with vertices A (0, 4), B (−2, 2) and C (6, 2)
Draw a triangle with vertices A (0, 4), B (−2, 2) and C (6, 2). Sketch the medians. Find the coordinates of the centroid. Step 2: Find the equation of another median A B C
𝑀 𝐴𝐶 = ,
𝑀 𝐴𝐶 = 6 2 , 6 2
𝑀 𝐴𝐶 = 3, 3
𝑚 𝐵𝑀 = 3 − − −2
𝑚 𝐵𝑀 = 1 5
𝑦= 1 5 𝑥+𝑏
3 = 𝑏
3= 3 5 +𝑏
3− 3 5 =𝑏
𝑏= or 12 5
𝑦=− 1 5 𝑥+ 12 5

8. Draw a triangle with vertices A (0, 4), B (−2, 2) and C (6, 2)
Draw a triangle with vertices A (0, 4), B (−2, 2) and C (6, 2). Sketch the medians. Find the coordinates of the centroid.
Step 3: Use substitution or elimination to find the point of intersection
𝑦=− 1 7 𝑥 𝑦= 1 5 𝑥+ 12 5
35 − 1 7 𝑥 = 1 5 𝑥+ 12 5
− 35 7 𝑥 = 35 5 𝑥
−5𝑥+100=7𝑥+84
−5𝑥−7𝑥=84−100
−12𝑥=−16
−12𝑥 −12 = −16 −12
𝑥= (or 1.33) A B C 𝑦= 𝑦= 𝑦=
𝑦= 40 15
𝑦= (or 2.67)
the POI and the centroid is , 40 15

9. Circumcentre
The point of intersection of the three perpendicular bisectors of a triangle

10. Draw a triangle with vertices A (0, 4), B (−2, 2) and C (6, 2)
Draw a triangle with vertices A (0, 4), B (−2, 2) and C (6, 2). Sketch the perpendicular bisectors. Find the coordinates of the circumcentre. A B C

11. Draw a triangle with vertices A (0, 4), B (−2, 2) and C (6, 2)
Draw a triangle with vertices A (0, 4), B (−2, 2) and C (6, 2). Sketch the perpendicular bisectors. Find the coordinates of the circumcentre. Step 1: Find the equation of one of the perpendicular bisectors 𝑚 𝐵𝐶 = 2 − 2 6 − −2 𝑚 𝐵𝐶 = 0 8 or 0 ⊥ 𝑚 𝐵𝐶 =𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑 A B C
𝑀 𝐵𝐶 = − ,
𝑀 𝐵𝐶 = 4 2 , 4 2
𝑀 𝐵𝐶 =(2, 2)
𝑥=2

12. Draw a triangle with vertices A (0, 4), B (−2, 2) and C (6, 2)
Draw a triangle with vertices A (0, 4), B (−2, 2) and C (6, 2). Sketch the perpendicular bisectors. Find the coordinates of the circumcentre. Step 2: Find the equation of another perpendicular bisector 𝑚 𝐴𝐵 = 2 − 4 −2 − 0 𝑚 𝐴𝐵 = −2 −2 or 1 ⊥ 𝑚 𝐴𝐵 =−1 A B C
𝑀 𝐴𝐵 = −2 2 ,
𝑀 𝐴𝐵 = −2 2 , 6 2
𝑀 𝐴𝐵 =(−1, 3)
𝑦=−𝑥+𝑏
3 =− −1 +𝑏
3=1+𝑏
3−1=𝑏
𝑏=2
𝑦=−𝑥+2

13. Draw a triangle with vertices A (0, 4), B (−2, 2) and C (6, 2)
Draw a triangle with vertices A (0, 4), B (−2, 2) and C (6, 2). Sketch the perpendicular bisectors. Find the coordinates of the circumcentre. Step 3: Use substitution or elimination to find the point of intersection 𝑥=2 𝑦=−𝑥+2 𝑦=− 2 +2 𝑦=0 A B C
the POI and the circumcentre is (2, 0)

14. Centroid The point of intersection of the three medians of a triangle
Calculate the equations of two of the medians
Solve using substitution

15. Circumcentre
The point of intersection of the three perpendicular bisectors of a triangle
Calculate the equations of two of the perpendicular bisectors
Solve using substitution

16. Success Criteria
I CAN find the centroid using the equations of two medians
I CAN find the circumcentre using the equations of two perpendicular bisectors

17. To Do…
Worksheet
Check the website daily for updates, missed notes, assignment solutions
New: note outline available the night before (completed note will no longer be posted)