Question

select the correct answer.
in the given figure, ( mwidehat{hj} = 106^circ ) and ( widehat{fh} cong widehat{jh} ). which statement is true?
figure not drawn to scale
a. the measure of ( angle g ) is ( 21^circ ), and triangle ( fgh ) is isosceles.
b. the measure of ( angle g ) is ( 56^circ ), and triangle ( fgh ) is isosceles.
c. the measure of ( angle g ) is ( 21^circ ), and triangle ( fgh ) is not isosceles.
d. the measure of ( angle g ) is ( 56^circ ), and triangle ( fgh ) is not isosceles.

Explanation:

Step1: Find arc $\overset{\frown}{FH}$

Since $\overset{\frown}{FH} = \overset{\frown}{JH}$, and the total circle is $360^\circ$. First calculate the remaining arc:
$360^\circ - 106^\circ = 254^\circ$
Then split equally: $\overset{\frown}{FH} = \frac{254^\circ}{2} = 127^\circ$

Step2: Find $\angle FHJ$ (inscribed angle)

The inscribed angle subtended by arc $\overset{\frown}{FH}$ is half the arc measure:
$\angle FHJ = \frac{1}{2} \times 127^\circ = 63.5^\circ$

Step3: Calculate $\angle GFH$

$\angle GFH$ is an inscribed angle subtended by $\overset{\frown}{HJ}$:
$\angle GFH = \frac{1}{2} \times 106^\circ = 53^\circ$

Step4: Find $\angle G$ using triangle exterior angle

$\angle FHJ$ is exterior to $\triangle FGH$, so $\angle FHJ = \angle G + \angle GFH$. Rearrange to solve for $\angle G$:
$\angle G = 63.5^\circ - 53^\circ = 10.5^\circ$
Correction: Use tangent-chord angle theorem
$\angle GFH$ is tangent-chord angle, equal to half the intercepted arc $\overset{\frown}{FH}$:
$\angle GFH = \frac{1}{2} \times 127^\circ = 63.5^\circ$
$\angle GHJ$ is tangent-chord angle, equal to half the intercepted arc $\overset{\frown}{HJ}$:
$\angle GHJ = \frac{1}{2} \times 106^\circ = 53^\circ$
$\angle FHG = 180^\circ - 63.5^\circ - 53^\circ = 63.5^\circ$
Since $\angle GFH = \angle FHG = 63.5^\circ$, $\triangle FGH$ is isosceles, and $\angle G = 180^\circ - 2\times63.5^\circ = 53^\circ$
Final correct method: Use arc relationships for tangent angles

1. $\overset{\frown}{FH} = \frac{360^\circ - 106^\circ}{2} = 127^\circ$
2. Tangent-chord angle $\angle GFH = \frac{1}{2}\overset{\frown}{FH} = 63.5^\circ$
3. Tangent-chord angle $\angle GHJ = \frac{1}{2}\overset{\frown}{HJ} = 53^\circ$
4. $\angle FHG = 180^\circ - 63.5^\circ - 53^\circ = 63.5^\circ$
5. $\angle G = 180^\circ - 63.5^\circ - 63.5^\circ = 53^\circ$

Correction to match options: Re-evaluate arc assignment
Given $\overset{\frown}{HJ}=106^\circ$, and $\overset{\frown}{FH}=\overset{\frown}{HJ}$, so $\overset{\frown}{FH}=106^\circ$. Then remaining arc $\overset{\frown}{FKJ}=360-2\times106=148^\circ$
$\angle G$ is formed by two tangents: $\angle G = \frac{1}{2}|\overset{\frown}{FKJ} - \overset{\frown}{FH}| = \frac{1}{2}|148-106|=21^\circ$
Since $GF=GH$ (tangents from a point to a circle are equal), $\triangle FGH$ is isosceles.

Answer:

A. The measure of $\angle$G is 21°, and triangle FGH is isosceles.