1. Straight Line Flashcards
2 forms of straight line equation
y = mx + c
(m is gradient, c is y intercept)
ax + by + c = 0
Gradient equation
m = y2 - y1 / x2 - x1
Gradient of line sloping up from left to right
+ive
Gradient of line sloping down from left to right
-ive
Gradient of a horizontal line
0
Gradient of a vertical line
Undefined
Lines with the same gradient are ______
parallel
Gradient of the line joining A (6,3) and B (3,8)
-5 / 3
Given the points C (2,p) and D (10,7) have a gradient of 1/4, find the value of p
p = 5
Requirements for collinearity
3 or more points on a straight line
Same gradient
Share a common point
3 or more points on a straight line
Same gradient
Share a common point
Requirements for collinearinity
Method for proving collinearity of A, B, and C
- Gradient AB
- Gradient BC
- Statement
Collinearity statement
Since mAB = mBC and share common point B, the points A, B and C are collinear
Are A (2,6), B (-1, 3) and C (3,8) collinear?
Not collinear as mAB ≠ mBC
tan0 =
Gradient, y2 - y1 / x2 - x1
What can be found with tan0 = m
If gradient is known, angle can be found
If angle is known, gradient can be found
If gradient is known, angle can be found
If angle is known, gradient can be found
m = tan0
Angle from triangle of straight line e =
Angle the straight line makes with x axis
Negative on bottom and top of fraction
Becomes positive as it cancels out
Calculate the size of the angle of the line joining P (3,9) and Q (-2, 1) (calc)
Gradient = 8 / 5
Angle = 57.99 °
How to find the distance between two known points
By constructing a right angle triangle
Distance formula
Find the distance between A (5,7) and B (-8, 2)
13.93
Rearrange 3x -2y = -7 and find a, b, and c
3x - 2y + 7 = 0
a = 3
b = -2
c = 7
rearrange y = -5x + 9 and find a b and c
5x + y - 9 = 0
a = 5, b = 1, c = -9
Rearrange to the general equation and find the values of a, b, and c:
y - 2 = 3/4 (x + 5)
-3x + 4y - 23 = 0
a = -3, b = 4, c = 23
find the gradient and y intercept:
4x + 2y + 10 = 0
m = -2, c = -5
find the gradient and y intercept:
3x -5y + 1 = 0
m = 3/5
c = 1/5
find the gradient and y intercept:
4 - x - 3y = 0
m = -1/3
c = 4/3
rearrange y = 1/4x - 1/3 into the form ax + by + c = 0
-3x + 12y + 4 =0
Equation for SL if we know one point and its gradient
y - b = m (x - a)
when to use y - b = m (x - a)
when you know one point and the gradient
components of straight line
Gradient
Point
Equation
Find the equation of the line passing through (4, 7) with the gradient of 2
y = 2x - 1
Find the equation of the line that passes through the point (4, 3) and (7, 15)
m = 4
a = 4
b = 3
y = 4x - 13
Perpendicular
Two lines that are at right angles to each other
Two lines that are at right angles to each other
Perpendicular
Rule for gradients of perpendicular lines
m1 x m2 = -1
To find a perpendicular gradient, flip the original gradient upside down and change the sign.
Find the perpendicular gradient of:
1/2
-5/3
0.25
1
-6
-2
3/5
-4
-1
1/6
0
Method for proving perpendicularity
- Find gradient of equation/line one
- Find gradient of equation/line two
- Statement - m1 x m2 = -1
Prove the lines x + 3y + 6 = 0 and y = 3x - 1 are perpendicular
Gradient 1 = -1/3
Gradient 2 = 3/1
Since m1 x m2 = -1, the lines are perpendicular
Find the equation of the line through the points (-3, 1) which is perpendicular to the line y = 2 - 4x
Gradient of y = 2 - 4x = -4
y = 1/4x + 7/4
Midpoint
Point exactly half way between two points
Point exactly half way between two points
midpoint
Midpoint equation
( x1 + x2 / 2, y1 + y2 / 2)
( x1 + x2 / 2, y1 + y2 / 2)
Midpoint
Find the midpoint of A (2, 8) and B (4, 6)
(3, 7)
Find the midpoint of (5, -7) and D (6, 4)
(5.5, -1.5)
Median
The line from a vertex to the midpoint of the opposite side. A triangle has 3 medians.
Vertex = corner of triangle
Where do 3 medians meet
Centroid
Find the equation of the median from L in the triangle with the vertices K (3, 7), L (-1, -5), and M (7, 3)
Mid point KM = (5, 5)
Gradient of L = 5/3
y = 5/3x - 10/3
Concurrent
Any number of lines that all pass through the same point
Altitude
A line that goes from a vertex to the opposite side and meets at a right angle
A line that goes from a vertex to the opposite side and meets at a right angle
Altitude
How many altitudes does a triangle have
3
What meet at the orthocentre
Altitudes
Method for finding equation of altitude
Find gradient of line perpendicular to altitude
Find gradient of altitude
Sub into y = m (x - a)
A is the point (2, -4), B (3, 1), C (-5, 0). Find the equation of the altitude from B
Gradient AC 4/-7
y = 7/4x - 17/4
Perpendicular bisector
A line that cuts another line in half, at right angles
How many perp. bisectors does a triangle have
3
What do perp. bisectors meet at
The circumcentre
Method for finding equation of perp. bisectors
- find midpoint of line perpendicular to perpendicular bisector
- Find gradient of line perpendicular to perpendicular bisector
- Find gradient of perpendicular bisector
- Sub known data into y = m (x - a)
Find perpendicular bisector of the line joining A (-4, 6) and C (-2, -2)
Mid point AC (-3, 2)
Gradient AC -4
Perpendicular gradient 1/4
y = 1/4x + 11/4
What do we use to find the point of intersection
Simultaneous equations
Method for simultaneous equations in points of intersection
Rearrange at least one of the equations into the form “x=“ or “y=“ and substitute into each other
OR
Write the equation in the form ax + by + c = 0 and cancel the x or y term
Find the point of intersection of the lines y = x + 1 and 4y + 2x = 16
(2, 3)
Find the point of intersection of 3x + 5y - 23 = 0 and 5x + 2y - 13 = 0
(1, 4)
Triangle PQR has the vertices P (-3, 5), Q (7, 3) and R (-1, -5). calculate the lengths of PQ, QR, and RP.
PQ = 10.2
QR = 11.31
RP = 10.2
