1. Grade 10 Academic (MPM2D) Unit 2: Analytic Geometry Altitude and Orthocentre
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
Altitude and Orthocentre
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3. 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

4. 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

5. 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
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6. 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
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7. 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
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8. 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
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9. 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
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10. 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
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11. 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
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12. 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

13. Continued: Area of Triangle ABC.
Recall: Area of Triangle:
B: (7, 2) C: (1, -6) D
Altitude and Orthocentre
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14. 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

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

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