Page 134 - Mathematics_Form_3

Circles

2. Measure the sizes of all the angles X
R
formed on the circumference of a
circle. Y

3. Compare the size of each angle
FOR ONLINE READING ONLY
measured in task 2 and conclude O

about the relationship between the
A B
angles.
4. Draw a similar figure in your
Figure 5.11: Angles in the same segment
exercise book and measure the
ˆ
In Figure 5.11, the angles AXB and
angles.
ˆ
AYB are formed in the same segment by
5. Compare the results in tasks 2 and the same arc AB. Therefore, according
ˆ
ˆ
4. to this theorem, AXB = AYB.
This theorem is proved as follows.
6. What is the general conclusion can
Consider Figure 5.12 as follows.
you give about the nature of the
X
angles formed by the same arc in
Y
the same segment?
7. Share your findings with other
O
students for further discussion.
B

A
In Activity 5.5, you have established a
relationship between the angles formed
Figure 5.12: Angles formed in a segment
by the same arc in the same segment.
Given a circle with centre O subtending
These angles are always equal. This ˆ ˆ
AXB and AYB at the circumference.
relationship is formally given in the ˆ ˆ
To prove AXB = AYB , proceed as
following theorem.
follows.
Construction: Join AO and BO as Mathematics for Secondary Schools
Theorem 5.2 shown in Figure 5.12.
Angles formed by the same arc in the Proof:
ˆ ˆ
ˆ
same segment of a circle are equal. This AOB 2ACB= AXB (angle at the centre)
theorem is referred to as “angles in the AOB 2ADB= AYB (angle at the centre)
ˆ
ˆ ˆ
same segment”.
ˆ
ˆ
2AXB = 2AYB
ˆ
ˆ
Theorem 5.2 is illustrated in Figure 5.11. Therefore, AXB = AYB .
Student\s Book Form Three 127

18/09/2025 09:59:39
MATHEMATIC F3 SB.indd 127 18/09/2025 09:59:39
MATHEMATIC F3 SB.indd 127