2.1 Length of a Line Segment

Transcription

1 2.1 Length of a Line Segment Consider the following line segment,, from ( 3,1) to (1,4). How could you find the length of? 4 B 2 A 5 5 Recall: 2 This leads us to the Distance Formula and Length of a Line Segment The distance between any two points (, ) and (, ) is: = ( ) + ( )

2 Examples: 1) Calculate the length of each line segment: a) (7,3), ( 4,1) b) ( 5,4), ( 3, 3) c) (0, 4), (6,7) We can also use the formula to classify triangles. 2) Classify the triangle as equilateral, isosceles, or scalene. (3,2) ( 3,4) (5,8)

3 2.2 Midpoint of a Line Segment Definition: When is the midpoint of a line segment with endpoints (, ) and (, ), the co-ordinates of are:, Examples: What is this really calculating? 1) Determine the co-ordinates of the midpoint,, of the line segment joining (3, 5) and (7,1). 2) Determine the midpoint and the length of the line segment joining ( 3,6) to ( 2,2). 3) In a triangle, a median is a line segment joining one vertex to the midpoint of the opposite side. A triangle has vertices ( 8,6), (4,10), (2 4). Calculate the lengths of the three medians.

4 2.3 Slope of a Line Segment Recall the slope formula for a line segment joining (, ) and (, ): = Examples: 1) Find the slope if the line segment joining each of the following: a) (2,5), (8,3) b) ( 4,7), (6,7) c) (10, 5), (3, 7) d) (4,9), (4, 17) 2) Determine if the following line segments are parallel, perpendicular, or neither. a) (2, 5), ( 2,1) (3, 1), ( 3,8) b) ( 2,3), (4,6) (6, 3), (2,5)

5 c) ( 8,7), (0, 5) ( 9,6), (3, 2)

6 2.4 Finding the Equation of a Line Recall that if you know the slope of a line and a point on that line, you can find the equation. Use the formula: = ( ) + to get an equation into = + form. Examples: 1) Determine the equation of the line that passes through ( 4,4) and (4,8). 2) Determine the equation of the line that passes through ( 3,6) and (2,4). 3) Write the equation from #2 in standard form. Now notice a new way you could do question #2:

7 2.4 Finding the Equation of a Line II Examples: Determine the equation of the line 1) that passes through (2, 4) and is parallel to = ) that passes through ( 3,6) and is perpendicular to = 0. 3) that passes through (1, 1) and has the same y-intercept as 2 5 = 10. 4) through the midpoint of XY where ( 6,4) and (4,2) and perpendicular to XY.

8 5) with an x-intercept of 5 and a y-intercept of -3 6) having an x-intercept of -3 and perpendicular to = 5.

9 2.5 Proving Properties of Triangles and Quadrilaterals Recall: Formulae that may be used: [Using A x 1, y ) and B x 2, y ) ] ( 1 ( 2 a) Length of a line segment AB = b) Midpoint of a line segment AB = c) Slope of a line segment AB = d) Finding the equation of a line given certain information Example: Given the vertices of a triangle as A ( 6,12), B(12, 12), C( 12,6) find: a) The equation of the median from A to BC A median of a triangle Solution: b) The equation of the altitude from B to AC An altitude of a triangle Solution:

10 c) The equation of the perpendicular bisector of BC A perpendicular bisector of a line segment Solution: EXTRA PRACTICE 1. Find the equation of the median from C to AB. 2. Find the intersection point of the two medians (from #1 and example). 3. Find the equation of the median from B to AC. 4. Is the intersection point found in #2 on the line found in #3? Explain how you decided this. Include a mathematical proof. 5. What conclusion can you reach about the medians of a triangle? Today: Homework pg. 104: #1-6 Tomorrow is an activity + Homework: pg. 104: #7-12

11 2.6 Equation of a Circle The point O(0,0) is called the. If we have a circle centered at the origin, the equation is: + = where (, ) represents any point on the circle and r represents the radius. Examples: 1) State the radius of each circle. a) + = 81 b) + = 169 c) + = ) Write the equation of each circle that is centered at the origin with a radius of: a) 6 b) 14 c) 3.7 3) Is the point (5, 2) on the circle + = 29? 4) How about ( 3,4)? 5) Determine the equation of a circle that is centered at the origin and passes through (8, 7)

12 A tangent is a line that passes through exactly one point on a circle. If that point is B on a circle centered at the origin, then the tangent through B is perpendicular to OB. 6) Find the equation of the circle centered at the origin and passing through (1, 4) and the equation of the tangent through that point. (Use standard form)

13 2.7 Equation of a Circle with Centre Exercise: Find the equation of a circle with centre ( 3,7) and radius 4. (Hint: Pythagorean Theorem) in general, the equation of a circle with centre (h, ) and radius r is: Examples: 1) State the centre and the radius of each circle: a) ( 3) + ( 5) = 49 b) ( + 7) + ( 3) = 121 2) Write the equation of the circle with: a) (5,3), = 4 b) ( 1, 7), = 12 3) Consider the circle ( 3) + ( + 5) = 100 a) Graph the circle

14 b) Determine if the points (10,2), (9, 13), ( 6,0) lie on the circle. 4) Given the circle ( 4) + ( + 1) = 65, state: a) the co-ordinates of the centre: b) the radius: c) the diameter: d) the intercepts: