Question

The circles below are concentric

What is the measure of the angle formed by the two secants?
What is the value of x?

Answer 1

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.

`$\Rightarrow \text{ angle} =(1)/(2)(121^\circ-57^\circ)`

`$\Rightarrow \text{ angle} =(1)/(2)(64^\circ)`

`$\Rightarrow \text{ angle} =32^\circ`

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.

`$\Rightarrow 32^\circ=(1)/(2)(x^\circ-78^\circ)`

Multiply by 2 on both sides.

`$\Rightarrow 2* 32^\circ= 2* (1)/(2)(x^\circ-78^\circ)`

`$\Rightarrow 64^\circ= x^\circ-78^\circ`

Add 78° on both sides.

`$\Rightarrow 64^\circ+78^\circ= x^\circ-78^\circ+78^\circ`

`$\Rightarrow 142^\circ= x^\circ`

The value of x is 124.