AS Test

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

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

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

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

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 = 0 + −2 2 , 4 + 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=2 6 7 or 20 7 𝑦=− 1 7 𝑥+ 20 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 = 0 + 6 2 , 4 + 2 2 𝑀 𝐴𝐶 = 6 2 , 6 2 𝑀 𝐴𝐶 = 3, 3 𝑚 𝐵𝑀 = 3 − 2 3 − −2 𝑚 𝐵𝑀 = 1 5 𝑦= 1 5 𝑥+𝑏 3 = 1 5 3 +𝑏 3= 3 5 +𝑏 3− 3 5 =𝑏 𝑏=2 2 5 or 12 5 𝑦=− 1 5 𝑥+ 12 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. Step 3: Use substitution or elimination to find the point of intersection 𝑦=− 1 7 𝑥+ 20 7 𝑦= 1 5 𝑥+ 12 5 35 − 1 7 𝑥+ 20 7 = 1 5 𝑥+ 12 5 − 35 7 𝑥+ 700 7 = 35 5 𝑥+ 420 5 −5𝑥+100=7𝑥+84 −5𝑥−7𝑥=84−100 −12𝑥=−16 −12𝑥 −12 = −16 −12 𝑥= 4 3 (or 1.33) A B C 𝑦= 1 5 4 3 + 12 5 𝑦= 4 15 + 12 5 𝑦= 4 15 + 36 15 𝑦= 40 15 𝑦= 8 3 (or 2.67) the POI and the centroid is 4 3 , 40 15

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

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

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 = −2 + 6 2 , 2 + 2 2 𝑀 𝐵𝐶 = 4 2 , 4 2 𝑀 𝐵𝐶 =(2, 2) 𝑥=2

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 = 0 + −2 2 , 4 + 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

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)

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

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

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

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