ANALYTICAL GEOMETRY
Analytical Geometry is the use of formulae to analyse the length, gradient and
midpoint etc. of certain points and their lines.
DISTANCE BETWEEN TWO POINTS
Remember the theorem of Pythagoras – it states that the hypotenuse squared is equal
to the sum of the squares of the other two sides.
To find the distance of any point:
GRADIENT
The gradient works out the slope of the graph – or the ratio of the height to the length.
CO-LINEAR POINTS
Three points are collinear when all three points lie on the same line and we check this
by using gradient, i.e. points A, B and C are collinear
Gradients along the straight line is the same throughout
WTS TUTORING 3
, PARALLEL AND PERPENDICULAR GRADIENTS
When one line is parallel to another line, the gradients are equal:
When one line is perpendicular (at a right angle) to another line, the gradients are the
negative inverse of each other or gradient 1 times gradient 2 is equal to negative
MIDPOINT
To find the midpoint of two coordinates you have to find the “average” of the two
coordinates and find the average of the two -coordinates.
&
MIDPOINT THEOREM:
If two midpoints on adjacent sides of a triangle are joined by a straight line, the
line will be parallel to and half the distance of the third side of the triangle.
WTS TUTORING 4
, EQUATIONS OF STRAIGHT LINES
KEY:
TWO COORDINATES
If you are given two coordinates work out the gradient, then substitute it into the
formula along with one of your coordinates and solve for c.
Finally, write down the equation.
GIVEN THE GRADIENT PARALLEL OR PERPENDICULAR TO ANOTHER LINE:
Determine if your gradient is the same (parallel lines) or the negative inverse of the
other gradient (perpendicular lines).
Then substitute your point to find c.
Finally write down your equation properly.
ANGLE OF INCLINATION
The angle of inclination tells you at what angle the straight line crosses the -axis.
To work out the angle of inclination first work out your gradient, then substitute it
into the formula and solve for the angle:
FORMULA:
Remember that when your gradient is positive you can simply use the angle given
when you work it out, however, if your gradient is NEGATIVE then work out the
angle using a positive value of the gradient then subtract the angle from 180°.
WTS TUTORING 5
, EQUATION OF A CIRCLE
Key notes:
The diameter is twice the radius
The radius is the same throughout the circle.
The tangent is perpendicular to the radius
A normal is a line perpendicular to the tangent at the point of contact – the
normal is not the radius but can go through the circle or be outside the circle.
A secant cuts the circle twice.
A chord touches the circle twice internally and divides the circle into segments
A sector is the middle piece between two radii.
A chord divides a circle’s circumference into different arcs.
A circumference is the distance around the circle.
CIRCLE WITH CENTRE AT THE ORIGIN
This formula should remind you of Pythagoras.
r is the radius and x and y is the coordinate at a point through the circle.
WTS TUTORING 6
Analytical Geometry is the use of formulae to analyse the length, gradient and
midpoint etc. of certain points and their lines.
DISTANCE BETWEEN TWO POINTS
Remember the theorem of Pythagoras – it states that the hypotenuse squared is equal
to the sum of the squares of the other two sides.
To find the distance of any point:
GRADIENT
The gradient works out the slope of the graph – or the ratio of the height to the length.
CO-LINEAR POINTS
Three points are collinear when all three points lie on the same line and we check this
by using gradient, i.e. points A, B and C are collinear
Gradients along the straight line is the same throughout
WTS TUTORING 3
, PARALLEL AND PERPENDICULAR GRADIENTS
When one line is parallel to another line, the gradients are equal:
When one line is perpendicular (at a right angle) to another line, the gradients are the
negative inverse of each other or gradient 1 times gradient 2 is equal to negative
MIDPOINT
To find the midpoint of two coordinates you have to find the “average” of the two
coordinates and find the average of the two -coordinates.
&
MIDPOINT THEOREM:
If two midpoints on adjacent sides of a triangle are joined by a straight line, the
line will be parallel to and half the distance of the third side of the triangle.
WTS TUTORING 4
, EQUATIONS OF STRAIGHT LINES
KEY:
TWO COORDINATES
If you are given two coordinates work out the gradient, then substitute it into the
formula along with one of your coordinates and solve for c.
Finally, write down the equation.
GIVEN THE GRADIENT PARALLEL OR PERPENDICULAR TO ANOTHER LINE:
Determine if your gradient is the same (parallel lines) or the negative inverse of the
other gradient (perpendicular lines).
Then substitute your point to find c.
Finally write down your equation properly.
ANGLE OF INCLINATION
The angle of inclination tells you at what angle the straight line crosses the -axis.
To work out the angle of inclination first work out your gradient, then substitute it
into the formula and solve for the angle:
FORMULA:
Remember that when your gradient is positive you can simply use the angle given
when you work it out, however, if your gradient is NEGATIVE then work out the
angle using a positive value of the gradient then subtract the angle from 180°.
WTS TUTORING 5
, EQUATION OF A CIRCLE
Key notes:
The diameter is twice the radius
The radius is the same throughout the circle.
The tangent is perpendicular to the radius
A normal is a line perpendicular to the tangent at the point of contact – the
normal is not the radius but can go through the circle or be outside the circle.
A secant cuts the circle twice.
A chord touches the circle twice internally and divides the circle into segments
A sector is the middle piece between two radii.
A chord divides a circle’s circumference into different arcs.
A circumference is the distance around the circle.
CIRCLE WITH CENTRE AT THE ORIGIN
This formula should remind you of Pythagoras.
r is the radius and x and y is the coordinate at a point through the circle.
WTS TUTORING 6
