Help with Central Angle and Inscribed Angles

Oct 2014
16
0
Memphis
Hello all. I am refreshing my geometry skills. I have a problem that states: Find the value of x. I know that
the central angle is equal to the length of the arc. I think what I have here is an inscribed angle but I do not know the value of the intercepted arc. I am lost. Please give me a hint to get me started. Thanks!
PracPicC5.gif
 

romsek

MHF Helper
Nov 2013
6,744
3,037
California
Draw the bisector of the $50^\circ$ angle to the center.

This bisector is also a radius and thus forms two isosceles triangles that are each $(25^\circ, 25^\circ, 130^\circ)$ (make sure this is clear)

Thus the angular measure of the circumference that is not subtended by $x$ is $2 \times 130^\circ = 260^\circ$

Clearly $x + 260^\circ = 360^\circ$ and thus $x=100^\circ$

You will need to prove that the bisector does in fact pass through the center of the circle. Use symmetry arguments.
 
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Dec 2012
1,145
502
Athens, OH, USA
Unfortunately, the "proof" provided by the link is fallacious. That "proof" requires that x + y is the measure of the inscribed angle PAQ. Tain't so.

 
Jul 2015
217
116
Ilford
Reflect Q in OA to the point Q′ on the other side of the circumference. Then $\angle\mathrm{PAQ^{\prime}}=x+y$.
 
Dec 2014
99
6
England
theorem, the angle which an arc of a circle subtends at the centre is double that which it subtends at the remaining part of the circumference,