In a circle, if two chords are congruent, then their corresponding central angles a...

In a circle, if two chords are congruent, then their corresponding central angles are congruent. The measure of chord GH is 11.4 units and the central angle subtended by GH is 66 degrees. Since chord GH is congruent to chord HJ, the central angle subtended by HJ is also 66 degrees. The length of a chord is given by $2r \sin(\theta/2)$, where r is the radius and $\theta$ is the central angle. In triangle OGH, OG and OH are radii, so $OG = OH$. Since $\angle GOH = 66^\circ$, triangle OGH is an isosceles triangle. We can find the radius using the law of cosines in triangle OGH: $GH^2 = OG^2 + OH^2 - 2(OG)(OH)\cos(66^\circ)$. Since $GH = 11.4$, we have $11.4^2 = r^2 + r^2 - 2r^2 \cos(66^\circ) = 2r^2(1 - \cos(66^\circ))$. So, $r^2 = \frac{11.4^2}{2(1 - \cos(66^\circ))}$. Now, for chord HJ, since $\angle HOJ = 66^\circ$ and OH and OJ are radii, triangle OHJ is also an isosceles triangle. The length of chord HJ is given by $HJ = 2r \sin(\angle HOJ/2)$. However, since GH and HJ are congruent chords, their lengths are equal. Therefore, HJ = GH = 11.4 units.