Given:
Intercepted arcs of small circle:
78° and x°
Intercepted arcs of large circle:
57° and 121°
To find:
The measure of angle formed by two secants and the value of x.
Solution:
Consider a large circle:
The angle made by two secants intersecting outside a circle is half the difference between the measure of intercepted arcs.
[tex]$\Rightarrow \text{ angle} =\frac{1}{2}(121^\circ-57^\circ)[/tex]
[tex]$\Rightarrow \text{ angle} =\frac{1}{2}(64^\circ)[/tex]
[tex]$\Rightarrow \text{ angle} =32^\circ[/tex]
The measure of the angle formed by two secants is 32°.
Two circles are concentric circles.
Therefore 32° is also the angle made by small circle arcs.
[tex]$\Rightarrow 32^\circ=\frac{1}{2}(x^\circ-78^\circ)[/tex]
Multiply by 2 on both sides.
[tex]$\Rightarrow 2\times 32^\circ= 2\times \frac{1}{2}(x^\circ-78^\circ)[/tex]
[tex]$\Rightarrow 64^\circ= x^\circ-78^\circ[/tex]
Add 78° on both sides.
[tex]$\Rightarrow 64^\circ+78^\circ= x^\circ-78^\circ+78^\circ[/tex]
[tex]$\Rightarrow 142^\circ= x^\circ[/tex]
The value of x is 124.
