This topic continues from Circles Part 1, which can be found here:

Circles P1 • Mathematics Revision Quiz | GCSE.CO.UK
Circles have unique characteristics that are essential to understand their relationships. These characteristics have specific terms associated with them that are important to recognise.
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In geometry, a circle is a 2-dimensional shape that consists of all the points equidistant from a central point called the centre. The distance from the centre to any point on the circle is called the radius. The distance across the circle passing through the centre is called the diameter, equal to twice the radius.

Here are some important terms related to circles:

Centre: The point from which all points on the circle are equidistant.
Radius: The distance from the centre to any point on the circle.
Diameter: The distance across the circle passing
Chord: A line segment joining any two points on the circle.
Tangent: A line that touches the circle at exactly one point.
Arc: A portion of the circle between two points on the circle.

Now let's look at some examples of how to use these terms in problems:



Example 1: Find the radius and diameter of a circle with a circumference of 20π units.

Solution: The formula for the circumference of a circle is C = 2πr, where r is the radius. We can rearrange this formula to solve for r:
C = 2πr 20π = 2πr r = 10

So the radius of the circle is 10 units. The diameter is twice the radius, so the diameter is 20 units.


Example 2: Find the length of a chord 8 units from the centre of a circle with a radius of 10 units.

Solution: Draw a diagram of the circle, with the centre labelled O, the chord labelled AB, and the point where the chord intersects the radius labelled M.

Circle Radius and Cord

Since OM is perpendicular to AB, we can use the Pythagorean theorem to find its length:

OM2 + MB2 = OB2 OM2 + (10-8)2 = 102 OM2 + 4 = 100 OM2 = 96 OM = 4√6

So, the length of the chord AB is twice OM or 8√6 units.


Example 3: A tangent to a circle with a radius of 6 units is drawn from a point 10 units away from the circle's centre. Find the length of the tangent.

Solution: Draw a diagram of the circle, with the centre labelled O, the tangent labelled AB, and the point where the tangent intersects the radius labelled M.

Since OM is perpendicular to AB, we can use the Pythagorean theorem to find its length:

OM2 + MB2 = OB2 OM2 + 62 = 102 OM2 = 64 OM = 8

Since AM is the circle's radius, it is also 6 units. So the length of the tangent AB is AM + MB, or 6 + 8 = 14 units.



Circles Revision Quiz

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