Circles

Angle Subtended by the Arc: The angle subtended by an arc is the angle formed by joining each end point of the arc to a point that lies on the opposite arc. In the figure given below, angle BAC is an angle subtended by arc BXC. If we join B and C, we also say that angle BAC is an angle subtended by chord BC.

A

C

Y

XB

Angles Inscribed in an Arc: Angle BAC is an angle inscribed in arc BXC.Intercepted Arc: If an angle and a circle is given, the arcs

that lie in the interior of the angle are said to be intercepted arcs. Intercepted arcs are shown.

AA

A

BBB CCC

XX X

Remember:- Angles subtended by the arc are congruent.- Angles inscribed in the same arc are congruent.- An angle inscribed in the semicircle is a right angle.

Example:A pole has to be erected on the boundary of a circular

park of diameter 13 m in such a way that the difference of its distance from two diametrically opposite fixed gates A and B on the boundary is 7 m. the distance of the pole from one of the gates is:

AA

BB

CC

Y

XX

(1) 8 meters (2) 8.25 meters(3) 5 meters (4) None of the above

Solution:Let P be the pole.As P lies on the circumference and AB is the

diameter, M angle APB = 90o Now, (x + 7)2 + x2 = 132 = 169Therefore, x2 + 7x – 60 = 0Therefore, (x + 12) (x - 5) = 0

P

A

x + 7 x

B13

Solving for x, we get, x = 5 and x + 7 = 12Hence, option 3.