Question

angle g is a circumscribed angle of circle e. major arc fd measures 280°. what is the measure of angle gfd? 40° 50° 80° 90°

Explanation:

Answer:

To find the measure of angle \( GFD \), we first note that the measure of a circumscribed angle (angle \( G \)) related to a circle can be found using the difference of the major and minor arcs. The total circumference of a circle is \( 360^\circ \), so the minor arc \( FD \) is \( 360^\circ - 280^\circ = 80^\circ \).

Now, triangle \( EFD \) is isosceles because \( EF \) and \( ED \) are radii of the circle (so \( EF = ED \)). The central angle \( \angle FED \) is equal to the measure of the minor arc \( FD \), which is \( 80^\circ \).

To find the base angles of triangle \( EFD \) (which are \( \angle EFD \) and \( \angle EDF \)), we use the fact that the sum of angles in a triangle is \( 180^\circ \). Let \( \angle EFD = \angle EDF = x \). Then:

\[
x + x + 80^\circ = 180^\circ
\]
\[
2x = 100^\circ
\]
\[
x = 50^\circ
\]

Wait, but angle \( GFD \) is related to the tangent and the chord. Wait, maybe I made a mistake. Let's re-examine.

Angle \( G \) is a circumscribed angle (tangent angle) formed by tangents \( GF \) and \( GD \) to circle \( E \). The measure of a circumscribed angle is half the difference of the measures of the intercepted arcs. The major arc \( FD \) is \( 280^\circ \), so the minor arc \( FD \) is \( 360 - 280 = 80^\circ \). Then the measure of angle \( G \) is \( \frac{1}{2}(280^\circ - 80^\circ) = \frac{1}{2}(200^\circ) = 100^\circ \).

Now, quadrilateral \( GFED \) has two right angles at \( F \) and \( D \) (since tangents are perpendicular to radii), so \( \angle GFE = \angle GDE = 90^\circ \). The sum of angles in a quadrilateral is \( 360^\circ \), so:

\[
\angle G + \angle GFE + \angle FED + \angle GDE = 360^\circ
\]
\[
100^\circ + 90^\circ + \angle FED + 90^\circ = 360^\circ
\]
\[
\angle FED = 80^\circ
\]

Now, in triangle \( EFD \), \( EF = ED \) (radii), so it's isosceles with \( \angle EFD = \angle EDF \). Then:

\[
\angle EFD = \frac{180^\circ - 80^\circ}{2} = 50^\circ
\]

But angle \( GFD \) is \( \angle GFE - \angle EFD \)? Wait, no. Wait, \( \angle GFE \) is \( 90^\circ \) (tangent perpendicular to radius), and \( \angle EFD \) is \( 50^\circ \), so \( \angle GFD = 90^\circ - 50^\circ = 40^\circ \)? Wait, that doesn't match. Wait, maybe I messed up.

Wait, let's start over. The measure of an inscribed angle is half the measure of its intercepted arc. But angle \( GFD \): what arc does it intercept? Wait, angle \( GFD \) is an inscribed angle? No, \( G \) is outside the circle, \( F \) and \( D \) are on the circle. Wait, maybe angle \( GFD \) is an inscribed angle intercepting arc \( GD \)? No, maybe not. Wait, the problem is asking for angle \( GFD \). Let's look at the diagram: \( G \) is outside, \( F \) and \( D \) are on the circle, \( E \) is the center. So \( GF \) and \( GD \) are tangents, so \( GF = GD \) (tangents from a common external point are equal), and \( EF \perp GF \), \( ED \perp GD \).

The major arc \( FD \) is \( 280^\circ \), so the minor arc \( FD \) is \( 80^\circ \). The central angle \( \angle FED \) is \( 80^\circ \). Now, in triangle \( EFD \), \( EF = ED \), so \( \angle EFD = \angle EDF = \frac{180 - 80}{2} = 50^\circ \). Now, \( \angle GFE = 90^\circ \) (tangent perpendicular to radius), so \( \angle GFD = \angle GFE - \angle EFD = 90^\circ - 50^\circ = 40^\circ \). Wait, but the options include \( 40^\circ \), \( 50^\circ \), \( 80^\circ \), \( 90^\circ \). So the answer should be \( 40^\circ \)? Wait, no, maybe I made a mistake. Wait, maybe angle \( GFD \) is an inscribed angle intercepting arc \( GD \)? No, let's check again.

Wait, the measure of angle \( GFD \): since \( GF \) is tangent to the circle at \( F \), the angle between tangent \( GF \) and chord \( FD \) is equal to half the measure of the intercepted arc \( FD \). Wait, the angle between a tangent and a chord is equal to half the measure of the intercepted arc. So angle \( GFD \) is the angle between tangent \( GF \) and chord \( FD \), so it should be half the measure of the intercepted arc \( FD \) (the minor arc). The minor arc \( FD \) is \( 80^\circ \), so half of that is \( 40^\circ \). Yes, that's the theorem: the measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc. So angle \( GFD \) is formed by tangent \( GF \) and chord \( FD \), so it's half the measure of arc \( FD \) (minor arc). Minor arc \( FD \) is \( 360 - 280 = 80^\circ \), so half of that is \( 40^\circ \). So the measure of angle \( GFD \) is \( 40^\circ \).

So the answer is \( 40^\circ \), which is the first option.