discuss the keywords in question has the same gradient as…
TRANSCRIPT
Discuss the keywords in question
Has the same gradient as…
Perpendicular and Parallel Lines
• Know what the terms parallel and perpendicular mean (especially in relation to straight lines). Know what a perpendicular bisector is.
• Understand the connection between the gradients for lines that are parallel and lines that are perpendicular.
• Be able to explain how to find the equations of lines that are perpendicular to another line or pair of points. Also be able to find the equation of the perpendicular bisector of two points.
Parallel lines
Parallel lines have the same gradient m
If asked for a parallel line to y = mx + c going through the point (x1, y1) you treat this like one point & with gradient.
Sub. m and x1 & y1 into 11 xxmyy
21 mm
Example
• Find the line parallel to y = 2x + 5, that passes through (2,7)
11 xxmyy 227 xy
32 xy
32 xy
Parallel lines
a)
b)3
14
3
2 xy
1432 yx
7,3 11 xxmyy
33
27 xy 9
3
2 xy
Perpendicular Lines
Perp. Lines do not have the same gradient
They have a negative reciprocal. What?
This just means this:
1
1
mmperp 1
2
1
mm
Perp. Lines
When asked to find a line perp. to another and going through the point (x1, y1)
Find m2 first. TIP: Change sign, and flip it.
3
41 m
4
32 m 51 m
5
12 m
Perp. Lines
Once you have found m2 ,then use line equation.
Sub. m and x1 & y1 into
11 xxmyy
Example
• Find the line perp. to y = 2x + 5, that passes through (2,7)
11 xxmyy 22
17 xy
82
1 xy
21 m2
12 m
a)
b)
1153 yx
5
11
5
3 xy
5
31 m
3
52 m 11 xxmyy
1,2 23
51 xy
3
7
3
5 xy
Try a k question
• A line that is perpendicular to the line that connects A(5,7) and B(7,-9) passes through the point (6,-1) and (-2, k) find the value of k
• k=-2 857
791
m8
12 m
62
1
8
1
k
8
1
8
1
k
11 k
Right angles
You might be asked to prove two lines crossing form a right angle between them (a.k.a that they are perp. To each other)
The proof is simple: 121 mm
Right angles
• The points A,B and C are (2,5); (5,9) and (6,2) respectively.
• Find the gradient of AB
• Find the gradient of AC and hence state whether the angle BAC is 90o
3
4
25
591
m
4
3
26
522
m 14
3
3
4
Try one out• A(6,4); B(9,8); C(10,0)• Find the gradients of AB; AC and BC• Decide if ABC is a right-angled triangle.• Gradients AB 4/3 AC1 BC 8 • None of the gradients are negative reciprocals
to each other so it is not a right-angled triangle.
Independent Study
Exercise 1F p23 (solutions p416)