3.2.2 Circles II, PT3 Practice

Posted on January 7, 2018 by user

3.2.2 Circles II, PT3 Practice

Question 6:

Diagram below shows a circle with centre O. POR is a straight line.

Find the value of x and of y.

Solution:

$\begin{array}{l} x = 40 \times 2 \\ = 80 \\ y = \frac{180 - 80}{2} \\ = 50 \end{array}$

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Question 7:
Diagram below shows a circle with centre O.

Find the value of x.

Solution:

$\begin{array}{l} ∠ Q R O = 38^{o}, ∠ O R S = 44^{o} \\ ∠ Q R S = 38^{o} + 44^{o} = 82^{o} \\ x^{o} = 82^{o} \times 2 \\ x^{o} = 164^{o} \\ x = 164 \end{array}$

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Question 8:
Diagram below shows a circle. QTS is the diameter of the circle and PTR is a straight line.

Find the value of x.

Solution:

$\begin{array}{l} ∠ R P S = 90^{o} - 42^{o} = 48^{o} \\ x^{o} = 48^{o} + 37^{o} = 85^{o} \end{array}$

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Question 9:
Diagram below shows a circle with centre O. PQR is a straight line.

Find the value of x and of y.

Solution:
In cyclic quadrilateral,
Angle at Q = 180^o – 70^o = 110^o
x² = exterior angle at Q
= 180^o – 110^o
= 70^o
x = 70

y^o = reflex angle at O
y^o = 360^o – (70^o × 2)
y^o = 220^o
y = 220

Question 10:
In diagram below, ABCD and DEFG are straight lines.

Find the value of x and of y.

Solution:

$\begin{array}{l} x = 95 \\ (exterior of ∠ B cyclic quadrilateral) \\ ∠ C B F + ∠ B F E + 95^{o} + 110^{o} = 360^{o} \\ ∠ C B F + ∠ B F E + 205^{o} = 360^{o} \\ ∠ C B F + ∠ B F E = 155^{o} \\ y^{o} + ∠ C B F + ∠ B F E = 180^{o} \\ y^{o} + 155^{o} = 180^{o} \\ y = 25 \end{array}$

3.2.1 Circles II, PT3 Practice

Posted on January 5, 2018 by user

3.2.1 Circles II, PT3 Practice

Question 1:

(a) In Diagram below, KLM, LPN and MPO are straight lines.

Find the value of x and of y.

(b) In the diagram below, O is the centre of the circle. Find the value of

(i) x (ii) y

Solution:

(a)

$\begin{array}{l} ∠ L O M = ∠ L N M = y^{o} = 44^{o} \\ ∴ y = 44 \\ ∠ N M O = ∠ N L O = 41^{o} \\ x^{o} = 180^{o} - 80^{o} - 41^{o} \\ = 59^{o} \end{array}$

(b)(i)

2x = 200^o

x = 100^o

(b)(ii)

x + y = 180^o

100^o + y = 180^o

y = 80^o

Question 2:

(a) In Diagram below, AOC and BCD are straight lines with centre O.

Find the value of y.

(b) In diagram below, ABCD is a circle with centre O and ADE is a straight line.

Find the value of y.

Solution:

(a)

$\begin{array}{l} ∠ A O B = 45^{o} \times 2 = 90^{o} \\ y^{o} = 180^{o} - 90^{o} = 90^{o} \end{array}$

(b)

$\begin{array}{l} ∠ A B D = \frac{1}{2} \times 100^{o} = 50^{o} \\ ∠ C D E = ∠ A B C = 85^{o} \\ y^{o} + 50^{o} = 85^{o} \\ y^{o} = 35^{o} \\ y = 35 \end{array}$

Question 3:

(a) In diagram below, ABCD is a circle. AEC and BED are straight lines.

Find the value of y.

(b) Diagram below shows a circle KLMN with centre O.

Find the value of x.

Solution:

(a)

$\begin{array}{l} ∠ A B E = ∠ A C D = 45^{o} \\ y^{o} = 180^{o} - 45^{o} - 55^{o} \\ y^{o} = 80^{o} \\ y = 80 \end{array}$

(b)

$\begin{array}{l} ∠ O M L = ∠ O L M = 42^{o} \\ (x^{o} + 42^{o}) + ∠ K N M = 180^{o} \\ x^{o} + 42^{o} + 110^{o} = 180^{o} \\ x^{o} + 152^{o} = 180^{o} \\ x^{o} = 180^{o} - 152^{o} \\ x^{o} = 28^{o} \\ x = 28 \end{array}$

Question 4:

Diagram below shows a semicircle ABCD with centre O.