6.6.1 Angles and Tangents of Circles, PT3 Practice

6.6.1 Angles and Tangents of Circles, 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}