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

Altitude and Orthocentre shows the height of a polygon to draw an altitude  line from vertex to opposite side so that it meets at 90° Orthocentre: where all 3 altitudes meet Altitude and Orthocentre © 2017 E. Choi – MPM2D - All Rights Reserved

Instructions Altitude and Orthocentre Plot the following points on the Cartesian plane given below. A(-1, 4) B(7, 2) C(1, -6) b) Determine the slope of side BC. Draw a line perpendicular to BC through A. What is this line called? Determine the slope and equation of this line. Repeat steps (b) to (e) for the other two altitudes. Prove that the altitudes of a triangle pass through the same point. What is that point called? Find point D and the length of the Altitude AD. Altitude and Orthocentre © 2017 E. Choi – MPM2D - All Rights Reserved

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

Continued b) Determine the slope of side BC. Draw a line perpendicular to BC through A. What is this line called? Determine the slope and equation of this line. Altitude Equation of Altitude from A From A (-1,4) Negative reciprocal Altitude and Orthocentre © 2017 E. Choi – MPM2D - All Rights Reserved

Continued Repeat steps b to e for the other 2 altitudes from B and C. Equation of Altitude from B From B (7, 2) Negative reciprocal Altitude and Orthocentre © 2017 E. Choi – MPM2D - All Rights Reserved

Continued Repeat steps b to e for the other 2 altitudes from B and C. Equation of Altitude from C From C (1, -6) Negative reciprocal Altitude and Orthocentre © 2017 E. Choi – MPM2D - All Rights Reserved

Continued: Point of Intersection (Orthocentre) Prove that the altitudes 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 altitudes intersect at the same point. Recall: Let (1) = (2) Altitude and Orthocentre © 2017 E. Choi – MPM2D - All Rights Reserved

Continued: Point of Intersection (Circumcentre) Prove that the altitudes of a triangle pass through the same point. Check by substitute Recall: Therefore all 3 altitudes intersect at the same point  Orthocentre: Altitude and Orthocentre © 2017 E. Choi – MPM2D - All Rights Reserved

Continued: Length of Altitude To find Point D Point D can be found by letting Line BC = Altitude AD Line BC: D Altitude and Orthocentre © 2017 E. Choi – MPM2D - All Rights Reserved

Continued: Length of Altitude Length of Altitude AD . Recall: Distance between 2 points: A: (-1 4) D: (5.08, -0.56) D You can use the similar techniques to find the other 2 altitudes. Altitude and Orthocentre © 2017 E. Choi – MPM2D - All Rights Reserved

Continued: Area of Triangle ABC . Recall: Area of Triangle: B: (7, 2) C: (1, -6) D Altitude and Orthocentre © 2017 E. Choi – MPM2D - All Rights Reserved

Altitude and Orthocentre shows the height of a polygon to draw an altitude  line from vertex to opposite side so that it meets at 90° Orthocentre: where all 3 altitudes meet Altitude and Orthocentre © 2017 E. Choi – MPM2D - All Rights Reserved

Homework Work sheet: Extra practice #1a-d Text: Check the website for updates Altitude and Orthocentre © 2017 E. Choi – MPM2D - All Rights Reserved

End of lesson Altitude and Orthocentre © 2017 E. Choi – MPM2D - All Rights Reserved