Circle Geometry Vocabulary and Properties for the TMUA

Basic Circle Vocabulary

To succeed in the TMUA, you must be fluent in conventional circle terminology. A circle consists of several distinct parts:

1. Centre: The fixed point from which all points on the circumference are equidistant.
2. Radius: The constant distance from the centre to any point on the circle.
3. Diameter: A straight line passing through the centre, connecting two points on the circumference. It is exactly twice the length of the radius.
4. Circumference: The perimeter or boundary of the circle.
5. Chord: A straight line segment joining any two points on the circumference.
6. Tangent: A straight line that touches the circumference at exactly one point and is perpendicular to the radius at that point.
7. Arc: A portion of the circumference. An arc smaller than a semicircle is a minor arc, while one larger is a major arc.
8. Sector: The region bounded by two radii and an arc (resembling a pizza slice). A minor sector has an angle less than 180 degrees, while a major sector is larger.
9. Segment: The region bounded by a chord and an arc. A minor segment is the smaller part created by the chord, and the major segment is the larger part.

The Equation of a Circle

Imagine drawing a circle of radius 1 on the $xy$ plane with its centre at the origin. All points on this circle are a distance of 1 from the origin. Using Pythagoras' theorem, we can express this as $x^2 + y^2 = 1^2$. Any $(x, y)$ satisfying this equation is on the circle.

If we increase the radius to $r$, the equation becomes $x^2 + y^2 = r^2$.

To move the centre to a different point $(a, b)$, we use a translation. Pythagoras' theorem tells us that the distance between $(x, y)$ and $(a, b)$ is $r$, leading to the standard form:

$(x - a)^2 + (y - b)^2 = r^2$

Identifying Centre and Radius

You must be able to identify the radius and centre quickly. For example, $(x - 2)^2 + (y - 3)^2 = 25$ has a centre at $(2, 3)$ and a radius of 5 (since $5^2 = 25$). Circles are often written in the expanded form $x^2 + y^2 + cx + dy + e = 0$. To find the centre and radius from this form, you must complete the square for both $x$ and $y$.

Example: Finding the centre and radius

Find the centre and radius of $x^2 + y^2 + 4x + 2y - 12 = 0$.

1. Collect $x$ and $y$ terms: $x^2 + 4x + y^2 + 2y - 12 = 0$.
2. Complete the square: $(x + 2)^2 - 4 + (y + 1)^2 - 1 - 12 = 0$.
3. Rearrange: $(x + 2)^2 + (y + 1)^2 = 17$.

The centre is $(-2, -1)$ and the radius is $\sqrt{17}$.

Caution on non-circles

Not every equation of the form $x^2 + y^2 + ax + by + c = 0$ is a circle. For instance, $x^2 + y^2 - 4x - 6y + 20 = 0$ simplifies to $(x - 2)^2 + (y - 3)^2 = -7$. Since the sum of squares cannot be negative, no real $(x, y)$ values satisfy this, so it is not a circle.

Tangents and Intersections

A line can intersect a circle twice, once (tangent), or not at all.

Example: Finding a tangent

Find the values of $c$ for which $y = 2x + c$ is tangent to the circle $(x - 3)^2 + (y - 2)^2 = 9$.

1. Substitute the line into the circle: $(x - 3)^2 + (2x + c - 2)^2 = 9$.
2. Expand and rearrange into a quadratic in $x$: $5x^2 + 2(c - 7)x + (c - 2)^2 = 0$.
3. For a tangent, the discriminant must be zero ($b^2 - 4ac = 0$): $4(c - 7)^2 - 20(c - 2)^2 = 0$.
4. Solve for $c$: $c - 7 = \pm \sqrt{5}(c - 2)$. This gives two values for $c$, representing the two possible tangent lines.

Circle Properties and Theorems

You are expected to know and apply the following geometric properties:

1. Chords: The perpendicular from the centre to a chord bisects the chord.
2. Tangents: The tangent at any point is perpendicular to the radius at that point.
3. Angles at the Centre: The angle subtended by an arc at the centre is twice the angle subtended at the circumference.
4. Semicircles: The angle in a semicircle is a right angle.
5. Segments: Angles in the same segment are equal.
6. Cyclic Quadrilaterals: Opposite angles in a cyclic quadrilateral add to 180 degrees.
7. Alternate Segment Theorem: The angle between a tangent and a chord is equal to the angle in the alternate segment.

When solving problems, use angle chasing by looking for isosceles triangles formed by radii, or add auxiliary lines such as diameters or chords to reveal hidden relationships.