Test 2 Section 4 #36

OK, so here’s my version of the diagram:


The first thing you need to recognize is that because segments LM and MN are tangent to the circle, and segments LO and NO are radii, you have two right angles:


Furthermore, with those two right angles and the 60º angle you’re given in the question, you have three out of the four angles in a quadrilateral:


The angles in all quadrilaterals add up to 360º, so you can solve for the measure of angle LON.

360° = 90° + 60° + 90° + m∠LON
120° = m∠LON


So far so good? Now we just need to use the fact that a circle has 360° of arc, and we’re dealing with 120°, or \frac{1}{3} of that. If the whole circumference of the circle is 96, then…


\dfrac{120}{360}\times 96 = x


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