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03 Jan 2010, 10:04
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I'm going over the Circle explanations that Brunel has been so generous to supply and I'm having a difficult time with the Central Angle Theorem.
" An inscribed angle is exactly half the corresponding central angle. Hence, all inscribed angles that subtend the same arc are equal. Angles inscribed on the arc are supplementary. In particular, every inscribed angle that subtends a diameter is a right angle (since the central angle is 180 degrees). "
It's the "In particular" part where i get lost. Looking at the diagram provided (sorry i can't link it, but i have no idea how) am I to assume that alpha is 90 degrees and 2 alpha is 180?
A second question that I just wanted to double check on is the formula for finding an Angle=90L/PieR...Am I correct in assuming it's 90 and not 180 because it's coming from the perimeter of the circle and not the centre?
P.S Anyone who is looking to get a better understanding of the theory behind this stuff will be well served to down load the material provided by Brunel. Makes things a lot clearer.
" An inscribed angle is exactly half the corresponding central angle. Hence, all inscribed angles that subtend the same arc are equal. Angles inscribed on the arc are supplementary. In particular, every inscribed angle that subtends a diameter is a right angle (since the central angle is 180 degrees). "
It's the "In particular" part where i get lost. Looking at the diagram provided (sorry i can't link it, but i have no idea how) am I to assume that alpha is 90 degrees and 2 alpha is 180?
A second question that I just wanted to double check on is the formula for finding an Angle=90L/PieR...Am I correct in assuming it's 90 and not 180 because it's coming from the perimeter of the circle and not the centre?
P.S Anyone who is looking to get a better understanding of the theory behind this stuff will be well served to down load the material provided by Brunel. Makes things a lot clearer.
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03 Jan 2010, 12:28
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thanks
Yes, the central angle you refer to should be 180, and the inscribed 90. Many people have a hard time visualizing a diameter as an "angle" because after all, it is a straight line with no bend! But imagine a central angle of, say, 120 degrees. Now imagine that angle opening up (increasing) until it is completely open and the radii lie end-to-end on a single line. That is 180 degrees (like any line), and splits the circle into exact halves.
thanks
It sounds like it. The formula for arc length is usually given:
Arc Length = (Central Angle/360) * Circumference
We can manipulate as follows:
Arc Length = (Central Angle/360)*2pi*r = (Central Angle/180)*pi*r
So, Central Angle = (180*Arc Length)/(pi*r)
If Inscribed Angle = (1/2)* Central Angle, that implies that Inscribed Angle = (90*Arc Length)/(pi*r)
[Note: I had to solve it out this way to check it myself, as this is not how I have memorized the formula. Typically, the GMAT would have you solve for the arc given the angle, rather than solve for the angle given the arc.]
Archived Topic
Hi there,
This topic has been closed and archived due to inactivity or violation of community quality standards. No more replies are possible here.
Where to now? Join ongoing discussions on thousands of quality questions in our Quantitative Questions Forum
Still interested in this question? Check out the "Best Topics" block above for a better discussion on this exact question, as well as several more related questions.
Thank you for understanding, and happy exploring!
