Maths Tutorials

Co-ordinate Geometry

In geometry, a cartesian co-ordinate system in a plane is a co-ordinate system that specifies each point uniquely by a pair of real numbers called co-ordinates, which are the signed distances to the point from the two fixed perpendicular oriented lines, called co- ordinate lines, co-ordinate axes or just the axes of a system.

This is what a cartesian plane looks like

It is the basis of co-ordinate geometry and everything depends on this or has some connection to this

Co-ordinate geometry would not exist without this plane

This figure only shows a 2D plane with an x and y axis, but a 3D plane also exists with x, y and z axis

The horizontal line is called the x axis, and the vertical line is called the y axis. A point lying on the cartesian plane is demoted by its x and y co-ordinate enclosed in brackets i.e. (x, y)

The x value (abscissa) is measure from the origin (0,0) along the horizontal line and the y value is measured from the origin along the vertical line.

To calculate the distance between two points on the cartesian plane, the distance formula is used.

Let’s say 2 points are (x1, y1) and (x2, y2). The distance between the points or the length of the line connecting both points is given by:

Mid point and Section formula

If a line is connecting two points (x1, y1) and (x2, y2) on a cartesian plane, the midpoint of that line is given by:

Ex: Find the mid point of the line connecting (3, 5) and (7,3).
((3+7)/2, (5+3)/2) = (5, 4) is the midpoint

Ex: A point divides the line connecting (3, 4) and (5, 2) in the ratio 2:3. Find the point.
((2 x 5 + 3 x 3)/(2+3), (2 x 2 + 3 x 4)/(2+3)) = (19/5, 16/5)

Centroid

If 3 points on the Cartesian plane lie on a straight line, they are said to be co-linear. If the points don’t lie on a straight line, they aren’t co-linear and form a triangle. A median is a line drawn from one vertex of a triangle which bisects the opposite side. Triangles have 3 medians, drawn from each of the 3 vertices, and the meeting point of these medians is called the centroid of a triangle. If the 3 vertices of a triangle are the points (x1, y1), (x2, y2) and (x3, y3), the centroid is given by:

Centroid Formula

Centroid Formula Illustration

Ex: The 3 vertices of a triangle are (1, 3), (2, 4) and (3, 5). Calculate its centroid G.
G = ((1+2+3)/3, (3+4+5)/3)
G = (6/3, 12/3) = (2, 4)

As seen in the algebra unit, every straight line on the cartesian plane has a characteristic equation distinguishing it from other lines. They are equations in two variables.

They are in the form y = mx + c, where m is called the slope of the line and c is the y intercept of the line.

The slope of the line is the inclination of it towards the positive direction of the x axis and the x and y intercept are the points at which the line meets the x and y axis respectively. The point at which the line meets the x axis is always (x, 0) and the point at which the line meets the y axis is always (0, y).

In this figure: y = 2x + 3 is graphed on the plane The line meets the y axis at (0,3) hence its y intercept is 3 given by c.

The slope of the line is the co-efficient of x which is 2.

Its x intercept is -3/2 *Substitute y = 0 and solve for x

To calculate the equation of a line when its slope and a point (x₁, y₁) lying on it are given:

y − y₁ = m(x − x₁)

Example: Calculate the equation of a line with slope 2 and point (2, 7)
y − 7 = 2(x − 2) = > y − 7 = 2x − 4 = > y = 2x + 3

To calculate the equation of a line when its slope and a point (x₁, y₁) lying on it are given:

*(y2-y1)/(x2-x1) is the slope of the line

Example: Find the equation of a line containing points (3, 4) and (1, 2)
Y-4 = ((2-4)/(1-3))(x-3) => y - 4 = x-3 => y = x+1

The slope of the line can also be calculate using the tan function. If the angle of inclination between the line and the positive direction of the x axis is ‘a’, tan a will give the slope of the line

Area and Volume

Objects can be represented in either 2 dimensions, or 3 dimensions. The amount of space an object takes up in 2 dimensions is called area. The amount of space an object takes up in 3 dimensions is called volume.

Shapes like square, rectangle circle, and triangles are 2 dimensional. However, they can also be represented in 3 dimensions. A square is a cube in 3 dimensions, a triangle is a cone or a pyramid in 3 dimensions, a circle is a sphere, a rectangle is a cuboid, etc.

Surface area is the other area taken up by a 3 dimensional shape/object. 2D objects only have one side, hence surface area is not applicable for them. Surface area is applicable only for 3D objects like cubes, cuboids, spheres, cylinders, cones, pyramids etc.

Angle properties of shapes

Triangles

  • Sum of angles: The sum of angles in a triangle is 180°.

Quadrilaterals

  • Sum of angles: The sum of angles in a quadrilateral is 360°.
  • Types of quadrilaterals: Quadrilaterals are shapes with 4 sides, including squares, rectangles, rhombuses, parallelograms, and trapezium.
  • Parallelograms: Squares, rectangles, and rhombus are further categorized as parallelograms since their opposite sides are equal and parallel.

Diagonals of a parallelogram bisect each other, and in a rhombus and square, they bisect at right angles to each other.

Traditional Parallelogram

Rhombus

Triangles

Triangles are 3 sides shapes with an angle sum property of 180 degrees There are various types of triangles based on the length of their sides and their angles. Triangles in which all sides are equal are called equilateral. If 2 sides are equal, they are called isosceles. If all sides are unequal they are called scalene.

Congruency of Triangles :

Two triangles are said to be congruent if they are of the exact same size and shape. Their corresponding sides and angles are equal in length and measure.

Congruency of Triangles

Similarity of Triangles :

Two triangles are said to be similar if their corresponding angles are all equal and their sides bear a common ratio to each other.

Similarity of Triangles

Proving similarity of 2 triangles

Similarity of 2 triangles can be proven by either of these three theorems: AA (If 2 corresponding angles are equal, triangles are similar), SAS (If 2 corresponding sides are in proportion and one angle is equal in both triangles, they are similar) and SSS (if all 3 sides are proportionate to each other the triangles are similar).

Circles

A circle is a two dimensional shape where all points on the circumference are at the same distance from the center point.

The line joining two points on the circumference of the circle and passing through its center is called the diameter. Radius is half the length of the diameter.

A chord is a line joining two points on the circle. The largest chord is the diameter. All diameters are chords but not all chords are diameters.

Pi is the ratio of the circles circumference to its diameter. Circumference is 2 x pi x r

Pi is a constant 3.14159265.. which is approx 3.14

In the cartesian plane, a circle can be plotted with respect to the circle’s equation:

(x − h)² + (y − k)² = r² where (h, k) is the center of the circle and r is the radius of it
(x − 1)² + (y − 1)² = 4 signifies that the center of the circle is at (1,1) and its radius is 2

Properties of a circle

1.A line drawn from the radius of a circle to any chord bisects the chord and makes an angle of 90 degrees with it.

If the radius of the circle is known, the length of the chord can be calculated. In the diagram, the radius is r and the length of the line from the centre to the chord is h. It forms a right angled triangle.

Using Pythagoras theorem, half the length of the chord can be calculated. Then double that to get the length of the entire chord.

2.The angle in a semi circle is 90 degrees

3.If from a point outside the circle, a line is drawn to cut the circle, that line is called the tangent and the tangent makes an angle of 90 degrees with the radius

4. If from the same point, 2 tangents are drawn to a circle, the length of the tangents are equal

5. Angle subtended by an arc at the center of the circle is twice the angle subtended by the same arc anywhere else on the circle

Theorem 5

Theorem 6

Theorem 2

6. Angles in the same segment of a circle are equal

7. Opposite angles of a cyclic quadrilateral add up to 180 degrees

Problem

1. A circle has centre G, and points M and N lie on the circle. Line segments MH and NH are tangent to the circle at points M and N respectively. If radius of the circle is 168 millimeters and the perimeter of quadrilateral GMHN is 3856 millimeters what is the distance between G and H? Ans: 1768 millimeters

2. Calculate angle DAB from the figure. Ans: 63 degrees

Calculate angles OBD, BDO, DOB, OEB, EDF, and DFE. Ans: 13, 13, 154, 77, 26, 90

Parabolas and Ellipse

A parabola is a U shaped curve in which every point on the curve is equidistant from a point called the focus and a straight line called the directrix. A parabola is basically the graphical structure of a quadratic equation. A parabola can be either upward or downwards.

The graph on the left is a downward parabola and the one on
the right is an upward parabola.

The tip of a parabola is called the vertex A parabola is a quadratic equation given by :

  • 1. a x2 + bx + c
  • 2. a(x − h)(x − k)
  • 3. a(x − p)2 + q

In equation 3, (p, q) represents the point at which the vertex of a given parabola is at. In equation 2, h and k are the x intercepts of the parabola.

A parabola can be either upwards, downwards, towards the right or towards the left of the cartesian plane.

Ellipse :

An ellipse is a plane curve surrounding two focal points, such that for all points on the curve, the sum of the two distances to the focal points is a constant. It is ovular in shape.

Equation of an ellipse in the general form is:

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