Circle O is shown below. The diagram is not drawn to scale. If m∠R = 36°, what is m∠O?
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°.
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°.
The answer is 72° Could you explain the steps?
In a circle, an angle formed by a chord is half the measure of the intercepted arc. Since angle R intercepts arc O, which is the entire circumference of the circle, then angle R is half the measure of the entire circumference.
The formula for the circumference of a circle is C = 2πr. Since the entire circumference is intercepted by angle R, the measure of the entire circumference is 360°. Now, we can set up an equation to find the measure of angle O.
m∠R = 36°
360° = 2(36°) + m∠O
360° = 72° + m∠O
Subtracting 72° from both sides of the equation gives:
288° = m∠O
Therefore, the measure of angle O is 288°.
To find the measure of angle O, we need to use the properties of angles formed by intersecting lines and arcs on a circle.
In this case, we have an angle R that is formed by a tangent line and the radius of the circle. According to the tangent-chord angle theorem, the measure of angle R is equal to half the measure of its intercepted arc, which we will call arc RS.
Since m∠R = 36°, we know that arc RS must measure twice that because of the tangent-chord angle theorem. Therefore, m(arc RS) = 2 * m∠R = 2 * 36° = 72°.
Now, angle O is an inscribed angle that intercepts the same arc RS as angle R. According to the inscribed angle theorem, the measure of an inscribed angle is half the measure of its intercepted arc. Therefore, m∠O = 0.5 * m(arc RS) = 0.5 * 72° = 36°.
So, the measure of angle O is 36°.