Answer:
Correct answer is 4th option
[tex]m\angle EFG=83^\circ[/tex], [tex]m\angle GCE=97^\circ[/tex]
Step-by-step explanation:
Given:
There are two tangents on the circle C at the point E and G.
[tex]m\angle EFG=(3x+11)^\circ[/tex], and [tex]m\angle GCE=(5x-23)^\circ[/tex]
To find:
The value of central and circumscribed angles = ?
Solution:
First of all, let use recall a property of tangents on a circle.
The line joining the center of circle to the point on circle on which there is a tangent, make an angle of [tex]90^\circ[/tex] with the tangent itself.
i.e. [tex]\angle CGF = \angle CEF = 90^\circ[/tex] ([tex]\because[/tex] G and F are the points on circle's tangent drawn from point F.)
Now, we can see that CGFE is a quadrilateral.
And sum of all internal angles of a quadrilateral is equal to [tex]360^\circ[/tex]
[tex]m \angle C+m \angle G+m \angle F+m \angle E = 360^\circ\\\Rightarrow (5x-23)+90+3x+11+90=360\\\Rightarrow 8x-12+180=360\\\Rightarrow 8x-12=360-180\\\Rightarrow 8x=180+12\\\Rightarrow x=24^\circ[/tex]
[tex]m\angle EFG=(3x+11)^\circ\\\Rightarrow m\angle EFG=(3\times 24+11)^\circ = 83^\circ[/tex]
[tex]m\angle GCE=(5x-23)^\circ\\\Rightarrow m\angle GCE=(5\times 24-23)^\circ\\\Rightarrow m\angle GCE=120-23^\circ\\\Rightarrow m\angle GCE=97^\circ[/tex]
So, correct answer is 4th option
[tex]m\angle EFG=83^\circ[/tex], [tex]m\angle GCE=97^\circ[/tex]
