Video: US-SAT03S4-Q34-579170942169

The given figure shows a circle centered at the origin. If the arc π΅πΆπ· has a length of (27π β 4)/3, find the measure of β π΄ππ΅ in radians.

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Video Transcript

The given figure shows a circle centered at the origin. If the arc π΅πΆπ· has a length of 27π minus four over three, find the measure of angle π΄ππ΅ in radians.

This is the arc π΅πΆπ·. And this is the angle weβre interested in, π΄ππ΅. We know that the arc length is 27π minus four over three. But the information weβll need to solve this problem is the angle measure that creates this arc. We know that an arc length π  is equal to the angle that creates that arc times the radius. And so if we wanted to calculate this angle π, we would take the arc length and divide it by the radius. And because we know that the circle is centered at the origin and that πΆ is located at six, zero, we can say that the radius is six units. And that means π will be equal to the arc length 27π minus four divided by three, divided by the radius of six. We can rewrite that to say 27π minus four over three times one-sixth. Dividing by six is the same thing as multiplying by one over six. And so weβll multiply those denominators together and the numerators together. 27π minus four over 18 is the angle measure that creates the arc π΅πΆπ·.

First, we need to calculate a common denominator between the two fractions. If we multiply three π over two times nine over nine, these two fractions will have a common denominator of 18. Nine times three π equals 27π. And nine times two equals 18 minus 27π minus four over 18. Now that they have a common denominator, we can add and subtract in the numerator. Look carefully at the numerator. It is now 27π minus 27π plus four. We were subtracting negative four. And that became positive four. And then we can say that 27π minus 27π cancels out. Itβs equal to zero. So we have angle π΄ππ΅ will be equal to four over 18. And we wanna give it in its simplified form, which is two-ninths. Angle π΄ππ΅ measures two-ninths radians. Angle π, angle π΅ππ·, measures 27π minus four over 18 radians. Together, these two angles measure three π over two radians.

Weβre only interested in angle π΄ππ΅. And that is two-ninths radians.