Question

in the below figure, find the ( moverparen{hlk} ).
answer attempt 1 out of 2
additional solution no solution
( overparen{hlk}= )

Explanation:

Step1: Use the property of perpendicular chords

Since \(JM\perp HK\), the arcs \(JH\) and \(JK\) are related. Also, the sum of the measures of arcs in a circle is \(360^{\circ}\). But more importantly, since \(JM\perp HK\), we know that \((2x - 8)+(127 - x)=90\) (because the angle formed by the perpendicular chords gives a right - angle - related arc sum).

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This is wrong. Wait, actually, since \(JM\perp HK\), the sum of the measures of \(\overset{\frown}{JH}\) and \(\overset{\frown}{JK}\) is \(180^{\circ}\) (the sum of arcs intercepted by perpendicular chords from the center - related property, actually, if two chords are perpendicular and one is a diameter, but here, we use the fact that \((2x - 8)+(127 - x)=180\) (because the sum of arcs that form a semi - circle).

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Step2: Find the measure of \(\overset{\frown}{HLK}\)

First, find the measure of \(\overset{\frown}{JH}\): Substitute \(x = 61\) into \(2x-8\), we get \(2\times61-8=122 - 8=114^{\circ}\).
The measure of \(\overset{\frown}{HLK}\) is \(360-(114 + 90)\) (because \(\overset{\frown}{JM}\perp\overset{\frown}{HK}\), so the arc opposite to the right - angle part. Wait, another way: The sum of arcs in a circle is \(360^{\circ}\). The measure of \(\overset{\frown}{HLK}=360-(2x - 8)\)
Since \(x = 61\), then \(2x-8=114\). So \(m\overset{\frown}{HLK}=360 - 114=246^{\circ}\)

Answer:

\(246\)