## Alternate Segments

A segment is a region of a circle which is "cut off" from the rest of the circle by a secant or a chord.

The Alternate Segment theorem states:

*An angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.*

We can see this theorem applied in the diagram below, giving us to pairs of equal angles ($x$`x` and $y$`y`).

## Intersecting Chords

A chord is a line that goes from one point on the circle's circumference to another, without passing through the centre.

Intersecting chord theorem: when two chords intersect each other inside a circle, the products of their segments are equal. Using the diagram below, $CM\times DM=AM\times BM$`C``M`×`D``M`=`A``M`×`B``M`.

## Intersecting Secants

A secant is a line that touches and passes through two points on a circle's circumference. It can

Intersecting Secant Theorem: when two secant lines intersect each other outside a circle, the products of their segments are equal. Using this, in the diagram below, $AB\times AC=AD\times AE$`A``B`×`A``C`=`A``D`×`A``E`.

#### Worked Examples

##### Question 1

Calculate $x$`x`, giving reasons for your answer. Write your answer correct to 2 decimal places.

##### Question 2 - coming soon

Consider the figure.

a) Calculate $x$`x` (to the nearest two decimal places). Give reasons for your answer.

b) Calculate $y$`y` (to the nearest two decimal places). Give reasons for your answer.

c) Using your two previous answers, calculate $z$`z`. Give reasons for your answer.