Page 158 - Mathematics_Form_3

Circles

2. Draw two tangent lines from an Theorem 5.11
external point P. (a) Two tangents from a common

3. Label the points of contact of the external point are equal.
(b) The tangents subtend equal
tangents and the circle by R and S.
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angles at the centre.
4. Join ‾ OS , ‾ OR and ‾ OP . (c) The line joining the centre and
5. By using a ruler, measure the external point bisects the angle
lengths of the tangents ‾ PR and ‾ PS . between the tangents.

6. Compare the lengths of line Theorem 5.11 is described geometrically

segments PR and PS . as follows.

7. By using a protractor, measure In Figure 5.23, the lines TQ and TP
ˆ
ˆ
angles POR and POS and compare are tangents to a circle with centre O.
them. Since the two tangents are from the same
external point T, then ‾ TP = ‾ TQ .
ˆ
ˆ
8. Measure angles RPO and SPO and ˆ ˆ ˆ ˆ
compare them. QOT = POT, and QTO = PTO .
Q
9. Write a general statement to
summarize your findings.

10. Share your results with other O T
students for further discussion.

P
In Activity 5.14, you learned that, at a Figure 5.23: Tangents from a common
point of tangency, the tangent, and the external point

radius meet at a right angle. In Activity Theorem 5.11 can be proved as follows.
5.15, you have discovered that, two Consider Figure 5.23. Given a circle
tangents drawn from a common external centred at O with tangents TQ and Mathematics for Secondary Schools
point are congruent. These properties are TP.
useful especially in determining lengths To prove TP = TQ proceed as

of chords such as diameter and other follows.
chords related to circles. Construction: Join QO , PO , and TO .

Proof: In QOT∆ and POT∆ , it
implies that
QO = PO (radii of the same circle)

Student\s Book Form Three 151

18/09/2025 09:59:52
MATHEMATIC F3 SB.indd 151
MATHEMATIC F3 SB.indd 151 18/09/2025 09:59:52