Advertisements
Advertisements
Question
Prove that the angle in a segment shorter than a semicircle is greater than a right angle.
Advertisements
Solution
\[ \stackrel\frown{QP} \text{ is a major arc and } \angle PSQ \text{ is the angle formed by it in the alternate segment } . \]
\[ \text{ We know that the angle subtended by an arc at the centre is twice the angle subtended by it at any point of the alternate segment of the circle } . \]
`=> 2angle "PSQ" = "m"`
`=> 2angle "PSQ" = 360^circ - "m"`
`=> 2 angle"PSQ" = 360^circ - 180^circ ...(because angle "POQ" < 108^circ)`
`=> 2angle "PSQ" > 180^circ`
`=> angle "PSQ" > 90^circ`
Thus, the angle in a segment shorter than a semi-circle is greater than a right angle.
APPEARS IN
RELATED QUESTIONS
Fill in the blank:
Segment of a circle is the region between an arc and .................. of the circle.
Prove that a diameter of a circle which bisects a chord of the circle also bisects the angle subtended by the chord at the centre of the circle.
Given an arc of a circle, show how to complete the circle.
If O is the centre of the circle, find the value of x in the following figure
If O is the centre of the circle, find the value of x in the following figure
In the given figure, it is given that O is the centre of the circle and ∠AOC = 150°. Find ∠ABC.
Prove that the angle in a segment greater than a semi-circle is less than a right angle.
In the given figure, if ∠AOB = 80° and ∠ABC = 30°, then find ∠CAO.
If arcs AXB and CYD of a circle are congruent, find the ratio of AB and CD.
In the following figure, ∠ACB = 40º. Find ∠OAB.
