Since angle R is an inscribed angle that intercepts arc OT, which has a measure of 72° (since 2 x 72° = 144°, the total measure of the arc), we know that m∠R = 1/2(marc OT). Therefore, 36° = 1/2(marc OT). Solving for marc OT, we get marc OT = 72°.
Since angle OOT is a central angle that intercepts arc OT, which has a measure of 72°, we know that m∠OOT = marc OT = 72°.
Finally, since angle ROO is an exterior angle of triangle OOT, we know that m∠ROO = m∠R + m∠OOT = 36° + 72° = 108°.
Since angle ROO is a central angle that intercepts arc AOB, we also know that m∠ROO = 1/2(marc AOB). Therefore, 108° = 1/2(marc AOB). Solving for marc AOB, we get marc AOB = 216°.
Since arc AOB is the entire circumference of the circle, which has a measure of 360°, we know that the measure of angle O is equal to the measure of the remaining arc, which is 360° - 216° = 144°.
Therefore, m∠O = 144°.
Circle O is shown below. The diagram is not drawn to scale. If m∠R = 36°, what is m∠O?
4 answers
The answers wrong and could you please shorten the steps
Sure, here are shorter steps to solve the problem:
- Angle R intercepts arc OT with a measure of 72° since it's an inscribed angle. Therefore, m∠R = 1/2m(arc OT) = 36°.
- Angle OOT is a central angle and intercepts arc OT, so m∠OOT = m(arc OT) = 72°.
- Angle ROO is an exterior angle of triangle OOT, so m∠ROO = m∠R + m∠OOT = 36° + 72° = 108°.
- Angle ROO also intercepts arc AOB, so m∠ROO = 1/2m(arc AOB) = 108°.
- The measure of arc AOB is equal to the circumference of the circle which is 360°. Therefore, m∠O = m(arc AOB) - m∠ROO = 360° - 2(108°) = 144°.
Therefore, m∠O = 144°.
- Angle R intercepts arc OT with a measure of 72° since it's an inscribed angle. Therefore, m∠R = 1/2m(arc OT) = 36°.
- Angle OOT is a central angle and intercepts arc OT, so m∠OOT = m(arc OT) = 72°.
- Angle ROO is an exterior angle of triangle OOT, so m∠ROO = m∠R + m∠OOT = 36° + 72° = 108°.
- Angle ROO also intercepts arc AOB, so m∠ROO = 1/2m(arc AOB) = 108°.
- The measure of arc AOB is equal to the circumference of the circle which is 360°. Therefore, m∠O = m(arc AOB) - m∠ROO = 360° - 2(108°) = 144°.
Therefore, m∠O = 144°.
The answer is 72° Could you explain the steps?