Coordinate Geometry: Formulae, Definitions and Circle Theory Rules

Question 1

How do you calculate the midpoint of two points?

Answer 1

Coordinates
A(Xₐ, Yₐ)
B(Xᵦ, Yᵦ)
Midpoint Formula
The midpoint between points A and B is:
((Xₐ + Xᵦ) / 2, (Yₐ + Yᵦ) / 2)

Question 2

Define the term perpendicular bisecter

Answer 2

A line passing through the midpoint of the line segment AB, at 90°, is called the perpendicular bisector of AB.

Question 3

What should you remember about perpendicular lines?

Answer 3

Perpendicular lines form a 90° angle
Thus, to find out if two intersecting lines form a right angle we use the formula m₁ x m₂ = -1

Question 4

How to calculate the distance between two points?

Answer 4

√[(x₂ - x₁)² + (y₂ - y₁)²]

Question 5

What is the general equation of a circle?

Answer 5

General equation of a circle centre 0 (0,0) and radius:
(x )² + (y)² = r²

Question 6

What is the general equation of a circle with a centre C (a,b)?

Answer 6

General equation of a circle centre 0 (0,0) and radius:
(x-a )² + (y-b)² = r²

By expanding:
x² - 2ax + a² + y² - 2by + b² = r²
=> x² + y² - 2ax - 2by + a² + b² - r² = 0

• Centre: (a, b)
• Radius: √(a² + b² - r²)

Standard Form (SF):
x² + y² + 2ax + 2by + c = 0
Centre: (-a, -b)
Radius: √(a² + b² - c)

Example:
1. For x² + y² = 6:

• Centre: (0, 0)
• Radius: √6

2. For x² - 2y² = 6:
   Not a circle

Question 7

Question: What happens when a line segment is drawn from the center of a circle to the midpoint of a chord?

Answer 7

Answer: The line will intersect the chord at right angles (90°).

The perpendicular from the center to a chord:

Question 8

Question: At what angle does a radius meet a tangent to a circle?

Answer 8

Answer: A radius meets a tangent at 90° (a right angle).

A tangent and radius meet at right angles:

Question 9

Question: How do the lengths of two external tangents to a circle compare?

Answer 9

Answer: The two external tangents to a circle are of equal length.

Equal external tangents:

Question 10

Question: How does the angle subtended by an arc at the center compare to the angle subtended by the same arc at the circumference?

Answer 10

Answer: The angle at the center is twice the angle at the circumference.

Angle at the center is twice the angle at the circumference:

Question 11

Question: How do the angles subtended by an arc in the same segment compare?

Answer 11

Answer: The angles subtended in the same segment are equal.

Angles in the same segment are equal:

Question 12

Question: What is the angle subtended by a diameter at the circumference of a circle?

Answer 12

Answer: The angle subtended by a diameter at the circumference is a right angle (90°).

Angle in a semicircle is a right angle:

Question 13

Question: What is the sum of opposite angles in a cyclic quadrilateral?

Answer 13

Answer: The opposite angles of a cyclic quadrilateral add up to 180°.

Opposite angles of a cyclic quadrilateral add up to 180°:

Question 14

Question: What is the relationship between the angle between a tangent and a chord and the angle subtended in the alternate segment?

Answer 14

Answer: The angle between a tangent and a chord is equal to the angle subtended in the opposite segment.

Alternate segment theorem

Question 15

Question: What is the relationship between the segments of two intersecting chords inside a circle?

Answer 15

Answer: The product of the segments of one chord is equal to the product of the segments of the other chord:
𝐴𝑋⋅𝐵𝑋=𝐶𝑋⋅𝐷𝑋

Intersecting chords (internal)

Question 16

Question: What is the relationship between the lengths of the segments when two chords meet outside the circle?

Answer 16

Answer: The product of the external segment and the whole length of one chord is equal to the product of the external segment and the whole length of the other chord:
AX⋅BX=CX⋅DX.

Intersecting chords (external):