Question

Please help!! Study the diagram of circle C. A circumscribed angle, ∠EFG, is tangent to ⨀C at points E and G, and ∠ECG is a central angle. Point P lies on the major arc formed by points E and G. If m∠EFG=(3x+11)∘, and m∠GCE=(5x−23)∘, what are the measures of the central and circumscribed angles?

Answer 1

Correct answer is 4th option

`m\angle EFG=83^\circ`,
`m\angle GCE=97^\circ`

Explanation:

Given:

There are two tangents on the circle C at the point E and G.

`m\angle EFG=(3x+11)^\circ`, and
`m\angle GCE=(5x-23)^\circ`

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
`90^\circ` with the tangent itself.

i.e.
`\angle CGF = \angle CEF = 90^\circ` (
`\because` 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
`360^\circ`

`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`

`m\angle EFG=(3x+11)^\circ\\\Rightarrow m\angle EFG=(3* 24+11)^\circ = 83^\circ`

`m\angle GCE=(5x-23)^\circ\\\Rightarrow m\angle GCE=(5* 24-23)^\circ\\\Rightarrow m\angle GCE=120-23^\circ\\\Rightarrow m\angle GCE=97^\circ`

So, correct answer is 4th option

`m\angle EFG=83^\circ`,
`m\angle GCE=97^\circ`