MATHEMATICS; CIRCLE GEOMETRY

Circle Theorems

Some interesting things about angles and circles.

Inscribed Angle

First off, a definition:

Inscribed Angle: an angle made from points sitting on the circle's circumference.

A and C are "end points"
B is the "apex point"

Play with it here:

Drag a point!

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When you move point "B", what happens to the angle?

Inscribed Angle Theorems

An inscribed angle a° is half of the central angle 2a°

(Called the Angle at the Centre Theorem)

And (keeping the end points fixed) ...

... the angle a° is always the same,
no matter where it is on the same arc between end points:

Angle a° is the same.
(Called the Angles Subtended by Same Arc Theorem)

 

Example: What is the size of Angle POQ? (O is circle's centre)

Angle POQ = 2 × Angle PRQ = 2 × 62° = 124°

Example: What is the size of Angle CBX?

Angle ADB = 32° also equals Angle ACB.

And Angle ACB also equals Angle XCB.

So in triangle BXC we know Angle BXC = 85°, and Angle XCB = 32°

Now use angles of a triangle add to 180° :

Angle CBX + Angle BXC + Angle XCB = 180°

Angle CBX + 85° + 32° = 180°

Angle CBX = 63°

Angle in a Semicircle (Thales' Theorem)

An angle inscribed across a circle's diameter is always a right angle:

(The end points are either end of a circle's diameter,
the apex point can be anywhere on the circumference.)

Why? Because: The inscribed angle 90° is half of the central angle 180° (Using "Angle at the Centre Theorem" above)

 

Another Good Reason Why It Works

We could also rotate the shape around 180° to make a rectangle!

It is a rectangle, because all sides are parallel, and both diagonals are equal.

And so its internal angles are all right angles (90°).

 

So there we go! No matter where that angle is
on the circumference, it is always 90°

Example: What is the size of Angle BAC?

The Angle in the Semicircle Theorem tells us that Angle ACB = 90°

Now use angles of a triangle add to 180° to find Angle BAC:

Angle BAC + 55° + 90° = 180°

Angle BAC = 35°

 

Finding a Circle's Centre

We can use this idea to find a circle's centre:

• draw a right angle from anywhere on the circle's circumference, then draw the diameter where the two legs hit the circle
• do that again but for a different diameter

Where the diameters cross is the centre!

 

Cyclic Quadrilateral

A "Cyclic" Quadrilateral has every vertex on a circle's circumference:
A Cyclic Quadrilateral's opposite angles add to 180°: a + c = 180° b + d = 180°

Example: What is the size of Angle WXY?

Opposite angles of a cyclic quadrilateral add to 180°

Angle WZY + Angle WXY = 180°

69° + Angle WXY = 180°

Angle WXY = 111°

 

Tangent Angle A tangent line just touches a circle at one point. It always forms a right angle with the circle's radius. https://www.mathsisfun.com/geometry/circle-theorems.html