Page 220 - ICSE Math 8

Angle in a Semicircle

In the given figure, PQ is a diameter of the circle with centre at O and R is any point on
the circumference of the circle. Thus, ∠PRQ is an angle in the semicircle. Angle in a
semicircle is a right angle, i.e., ∠PRQ = 90°.
Example 6: In the given figure, PQ is the diameter of the circle and ∆PRQ is an
isosceles triangle with RP = RQ. Find ∠PQR.

Solution: Since PQ is the diameter of the circle, ∠PRQ = 90°.
∴ ∆PQR is a right-angled isosceles triangle with RP = RQ.
⇒ ∠RPQ = ∠RQP (Angles opposite to equal sides are equal)

In ∆PQR,
∠RPQ + ∠PQR + ∠PRQ = 180°
⇒ 2∠PQR + 90° = 180° ⇒ 2∠PQR = 180° – 90° ⇒ ∠PQR = 90° = 45°
2
∴ ∠PQR = 45°
Example 7: In the given figure, AB is the diameter of the circle with centre O and
∠ACB = 90°. If AC = 6 cm and AB = 10 cm, find BC.
Solution: In right-angled ∆ACB,

2
2
2
AB = AC + BC (Using Pythagoras theorem)
2
2
2
⇒ 10 = 6 + BC ⇒ 100 = 36 + BC 2
2
⇒ BC = 100 – 36 = 64 ⇒ BC = 8 cm
EXERCISE
1. Look at the given figure and fill in the blanks.

(a) Radii = ____________, ____________ and ____________
(b) Tangent = ____________
(c) Diameter = ____________
(d) Chord = ____________ and ____________
(e) Secant = ____________

2. Find the diameter of a circle with centre O and radius 8 cm. Also, find if points P, Q and R lie on the
circle, inside the circle or outside the circle if:
(a) OP = 6.5 cm (b) OQ = 8 cm (c) OR = 9 cm
3. In a circle of radius 5.5 cm, can we draw a chord of length 9 cm? Justify your answer.

4. Given that PQ is a tangent to a circle with P as the point of contact. If the radius of the circle with centre
O is 5 cm and OQ = 13 cm, find the length of PQ.
5. In the given figure, AB and AC are two tangents to the circle with centre O such
that ∠BOC = 110°. Find the measure of ∠BAC.
6. Given a circle with centre O and KL as its diameter, do the
following.
(a) Find ∠KNL.
(b) Is ∠KNL = ∠KML?
(c) If OL = 5 cm, ML = 6 cm, find KM.

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