Grade 10 Academic (MPM2D) Unit 2: Analytic Geometry Medians and Centroid Mr. Choi © 2017 E. Choi – MPM2D - All Rights Reserved

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 © 2017 E. Choi – MPM2D - All Rights Reserved

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

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

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

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 © 2017 E. Choi – MPM2D - All Rights Reserved

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 © 2017 E. Choi – MPM2D - All Rights Reserved

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

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

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

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

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

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

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

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