Alternate Segments and Intersecting Chords and Secants

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, $CM\times DM=AM\times BM$CM×DM=AM×BM.

 

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, $AB\times AC=AD\times AE$AB×AC=AD×AE.

 

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.