1. 21B: Circle theorems

2. Intersecting chords
Intersecting chords are lines that cross over each other within a circle
If two chords intersect like this, the following rule applies:

3. Example Find m On chord CD, we have values 6 and 5
On chord AB, we have values 4 and m
6 x 5 = 4 x m
30 = 4m
30 ÷ 4 = m
m = 7.5

4. Intersecting secants (1)
Remember, a secant is a chord that extends beyond the circle on one side
If two secants intersect as shown below (where the lines cut through the circle and touch the circumference on the opposite side), the following rule applies:

5. Example Find y These secants intersect at point X
Using our previous diagram:
a = = 12
b = 7
c = 6 + y
d = 6
Our rule states that a x b = c x d
12 x 7 = (6 + y) x 6
84 = y
84 – 36 = 6y
48 = 6y
48 ÷ 6 = y
y = 8

6. Intersecting secants (2)
If two secants intersect as shown below (where one secant touches the outside edge of the circle without cutting through the middle of it), the following rule applies:

7. Intersecting tangents
If two tangents intersect outside the circle, their lengths will be equal:
m = 3

8. Intersecting chords and radius (radii)
When a chord and radius intersect, the following rules apply:
1. If the radius makes a right angle with the chord, the chord is cut in half so that both sides of the chord are equal
2. If there are two chords of the same length which intersect with the radii making a right angle, then both the radii and chords are cut in half equally

9. Example Find the values of the pronumerals
In this example we have two chords which both make right angles with the radii
Therefore, length OF is the same as OE:
OF = 2.5, OE = n, therefore n = 2.5
As the radius makes a right angle with the chord, BE is the same length as AE:
EB = 3, AE = m, therefore m = 3

10. Questions to do
Exercise 21B page 713 questions