Question

prove all the theorams of circle?​

Answer 1

Step-by-step explanation:

Inscribed Angle

First off, a definition:

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

inscribed angle ABC

A and C are "end points"

B is the "apex point"

Play with it here:

Drag a point!© 2018 MathsIsFun.com v0.87

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

Inscribed Angle Theorems

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

inscribed angle a on circumference, 2a at center

(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:

inscribed angle alwyas a on circumference

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)

inscribed angle 62 on circumference

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

Example: What is the size of Angle CBX?

inscribed angle example

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:

angle inscribed across diameter is 90 degrees

(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)

angle semicircle 90 degrees and 180 at center

Another Good Reason Why It Works

angle semicircle rectangleangle semicircle rectangle

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°).

angle semicircle always 90 on circumference

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?

inscribed angle example

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

finding as circles center

Answer 2

Answer:

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Step-by-step explanation:

Circle theorems: where do they come from?

The angle at the centre is twice the angle at the circumference.

The angle in a semicircle is a right angle.

Angles in the same segment are equal.

Opposite angles in a cyclic quadrilateral sum to 180°

The angle between the chord and the tangent is equal to the angle in the alternate segment.