To provide the best experiences, we use technologies like cookies to store and/or access device information. Consenting to these technologies will allow us to process data such as browsing behavior or unique IDs on this site. Not consenting or withdrawing consent, may adversely affect certain features and functions.
The technical storage or access is strictly necessary for the legitimate purpose of enabling the use of a specific service explicitly requested by the subscriber or user, or for the sole purpose of carrying out the transmission of a communication over an electronic communications network.
The technical storage or access is necessary for the legitimate purpose of storing preferences that are not requested by the subscriber or user.
The technical storage or access that is used exclusively for statistical purposes.
The technical storage or access that is used exclusively for anonymous statistical purposes. Without a subpoena, voluntary compliance on the part of your Internet Service Provider, or additional records from a third party, information stored or retrieved for this purpose alone cannot usually be used to identify you.
The technical storage or access is required to create user profiles to send advertising, or to track the user on a website or across several websites for similar marketing purposes.
3 Responses
Quick tell: stand at B and “look across” the chord AC-the arc you see (the one not containing B) is the slice you double; if the inscribed angle is acute you’re doubling the minor arc, if it’s obtuse you’re doubling the major one. So with ∠ABC = 35° you should get the central angle on that same arc as 70°, and the 110° must be the other central angle measured the opposite way around the center (I suspect the diagram picked that), though I could be slightly off without your picture.
“Angle at the centre is twice the angle at the circumference” means: both angles must stand on the same arc. The inscribed angle ∠ABC intercepts the arc opposite B. A quick tell is to look through B across the chord AC and shade the arc that lies inside the opening of ∠ABC; that shaded arc is the one you double. Then pick the central angle ∠AOC whose rays OA and OC enclose exactly that shaded arc. If ∠ABC < 90°, the shaded arc is the minor arc and ∠AOC = 2∠ABC. If ∠ABC > 90°, the shaded arc is the major arc; the same-arc central angle is reflex with size 2∠ABC, and the smaller central angle is 360° − 2∠ABC.
In your example, ∠ABC = 35° intercepts the minor arc AC of 70°, so the central angle that stands on the same arc is 70°. A result of 110° would be using a different angle at O (one that does not stand on the same arc), or a different intended arc in the diagram. Under time pressure, shading the intercepted arc first removes the ambiguity.
Hope this helps!
I used to mix this up too! The “angle at the center is twice the angle at the circumference” always means: take the arc that the inscribed angle is looking at (the arc inside that angle), not the arc that contains the vertex on the rim. Quick tell: draw or imagine the two chords from the vertex B to A and C; the arc between A and C that lies inside ∠ABC is the one you double. If that arc is less than a semicircle (your inscribed angle is acute), the smaller central angle ∠AOC is just 2×∠ABC. If that arc is more than a semicircle (your inscribed angle is obtuse), the central angle on the same arc is reflex, equal to 2×∠ABC, so the smaller central angle between OA and OC is 360° − 2×∠ABC. In your example, ∠ABC = 35° is acute, so the intercepted arc is the short one and the correct central angle is 70°, not 110°. A simple worked example where 110° does appear: if ∠ABC = 125° (obtuse), then the arc inside that angle is the long arc AC, so the central angle on that same arc is 2×125° = 250° (reflex), and the smaller central angle is 360° − 250° = 110°.