Page 136 - Mathematics_Form_3

Circles

Find B
(a) The value of x.
ˆ
(b) The measures of PQS and
ˆ
PRS C
(c) The value of θ O
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5. In a circle with a centre at point O, A
a chord AB subtends an inscribed
angle ACB 35∠ = ° at point C on D
the circle. Figure 5.13: Concyclic points and cyclic
(a) Calculate the measure of an quadrilateral
angle AOB. Points A, B, C, and D in Figure 5.13 are
(b) If a point D is chosen on the concyclic points. By joining these points
same segment as C, show that in their order, a quadrilateral ABCD
∠ ADB also measures 35°. inscribed in the circle with centre at O
is formed. Quadrilaterals such as ABCD
6. In a circle centred at point O, whose vertices are points on the circle
a chord AB subtends inscribed are called cyclic quadrilaterals. Thus,
angles at points C, D, and E in ABCD is a cyclic quadrilateral. The
the same segment. Suppose the opposite angles in a cyclic quadrilateral
measures of two of the angles are are uniquely related as you will deduce
given by (2x + 10)° and (3x −°. in Activity 5.6.
5)
(a) Find the value of x.

(b) Determine the actual Activity 5.6: Deducing the
measure of the angles. relationship between opposite
7. ∆ ACD and BCD∆ are inscribed in a angles in a cyclic quadrilateral
circle whose diameter is BD and the
ˆ
6
chord CD. Given CAD 6y= −° Individually or in a group, perform
ˆ
and CBD 3y= + 12 , ° find the value the following tasks:
of y and measures of these angles. 1. Use a pair of compasses to draw Mathematics for Secondary Schools
a circle of convenient radius.
2. Form a cyclic quadrilateral using
Relationship between opposite angles
the circle you have drawn.
of a cyclic quadrilateral
If two or more points lie on the 3. Measure all four angles formed
in the cyclic quadrilateral.
circumference of a circle, they are said
to be concyclic. Figure 5.13 shows 4. Find the sum of the pairs of
concyclic points on a circle with centre opposite angles obtained in task
O. 3.

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MATHEMATIC F3 SB.indd 129