Page 162 - Mathematics_Form_3

Circles

7. On the other segment, draw two line Theorem 5.12 is described as follows.
segments each from one end of the In Figure 5.27, angle BAT is formed
chord and meet on the circle. between the chord AB and the tangent
ˆ
8. Label and measure the angle formed AT . Angle ACB is formed by an arc
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between the two line segments. which forms a segment with chord AB.
9. On the same circle drawn in task 1,
C
repeat tasks 7 and 8 several times,
each time the lines meet the circle
at different points.
B
10. By comparing the angles you
have measured, what unique
characteristics do you observe from
T
these angles? A

11. Write a general statement which Figure 5.27: An angle formed between
the chord and the tangent
summarizes your observation.
ˆ
12. Use the knowledge of algebra, Since the angle ACB is formed in the
congruence and similarity theorems, alternate (opposite) segment, Theorem
̂
̂
and the previous circle theorems to 5.12 implies that A C B = B A T.
prove what you have concluded in Theorem 5.12 can be proved geometrically
task 11. as follows.
13. Share your final work with other Consider a circle centred at O, AC is a
students for further discussion. tangent at B and BD is a chord as shown
in the following figure.

The conclusion you have drawn in F
Activity 5.16 is known as the angle in E
alternate segment of circle theorem.

D D
Theorem 5.12 Mathematics for Secondary Schools
In any circle, when a tangent at
point of contact intersects a chord
A B C
the angle formed between the chord
ˆ
ˆ
and tangent is equal to all angles To prove that DBC = BED, proceed
formed by the chord in the alternate as follows.
segment. Construction: Draw BF through the

centre and join DF .

Student\s Book Form Three 155

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MATHEMATIC F3 SB.indd 155 18/09/2025 09:59:54
MATHEMATIC F3 SB.indd 155