# 3.2.2 Circles II, PT3 Practice

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

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

Find the value of x.

Solution:
$\begin{array}{l}\angle QRO={38}^{o},\angle ORS={44}^{o}\\ \angle QRS={38}^{o}+{44}^{o}={82}^{o}\\ {x}^{o}={82}^{o}×2\\ {x}^{o}={164}^{o}\\ x=164\end{array}$

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:
$\begin{array}{l}\angle RPS={90}^{o}-{42}^{o}={48}^{o}\\ {x}^{o}={48}^{o}+{37}^{o}={85}^{o}\end{array}$

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

Find the value of x and of y.

Solution:
Angle at Q = 180o – 70o = 110o
x2 = exterior angle at Q
= 180o – 110o
= 70o
x = 70

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

Question 10:
In diagram below, ABCD and DEFG are straight lines.
Find the value of x and of y.

Solution:

# 3.2.1 Circles II, PT3 Practice

3.2.1 Circles II, PT3 Practice

Question 1:
(a) In Diagram below, KLMLPN 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)
$\begin{array}{l}\angle LOM=\angle LNM={y}^{o}={44}^{o}\\ \therefore y=44\\ \angle NMO=\angle NLO={41}^{o}\\ {x}^{o}={180}^{o}-{80}^{o}-{41}^{o}\\ \text{}={59}^{o}\end{array}$

(b)(i)
2x = 200o
x = 100o

(b)(ii)
+ y = 180o
100o + y = 180o
y = 80o

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)
$\begin{array}{l}\angle AOB={45}^{o}×2={90}^{o}\\ {y}^{o}={180}^{o}-{90}^{o}={90}^{o}\end{array}$

(b)

$\begin{array}{l}\angle ABD=\frac{1}{2}×{100}^{o}={50}^{o}\\ \angle CDE=\angle ABC={85}^{o}\\ {y}^{o}+{50}^{o}={85}^{o}\\ \text{}{y}^{o}={35}^{o}\\ \text{}y=35\end{array}$

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)
$\begin{array}{l}\angle ABE=\angle ACD={45}^{o}\\ {y}^{o}={180}^{o}-{45}^{o}-{55}^{o}\\ {y}^{o}={80}^{o}\\ y=80\end{array}$

(b)
$\begin{array}{l}\angle OML=\angle OLM={42}^{o}\\ \left({x}^{o}+{42}^{o}\right)+\angle KNM={180}^{o}\\ {x}^{o}+{42}^{o}+{110}^{o}={180}^{o}\\ {x}^{o}+{152}^{o}={180}^{o}\\ {x}^{o}={180}^{o}-{152}^{o}\\ {x}^{o}={28}^{o}\\ x=28\end{array}$

Question 4:
Diagram below shows a semicircle ABCD with centre O.

It is given that AB = CD.
Find the value of y.

Solution:
y+ 32o + 32o + 90o = 180o
y= 180o – 154o
y= 26o
= 26

Question 5:
In diagram below, ABC and DEF are straight lines.

Find the value of x and of y.

Solution:

# 3.1 Circles II

3.1 Circles II

(A) Properties of Circles Involving Symmetry, Chords and Arcs

(B) Identify the Properties of Chords
(1) A radius that is perpendicular to a chord divides the chord into two equal parts.

(C) Properties of Angles in Circles

(1)

(2)

(3)

(4)

(5)