Advertisements
Advertisements
प्रश्न
If two equal chords of a circle intersect within the circle, prove that the line joining the point of intersection to the centre makes equal angles with the chords.
उत्तर
Let PQ and RS are two equal chords of a given circle and they are intersecting each other at point T.
Draw perpendiculars OV and OU on these chords.
In ΔOVT and ΔOUT,
OV = OU ...(Equal chords of a circle are equidistant from the centre)
∠OVT = ∠OUT ....(Each 90°)
OT = OT ...(Common)
∴ ΔOVT ≅ ΔOUT ...(RHS congruence rule)
∴ ∠OTV = ∠OTU ...(By CPCT)
Therefore, it is proved that the line joining the point of intersection to the centre makes equal angles with the chords.
APPEARS IN
संबंधित प्रश्न
Two circles of radii 5 cm and 3 cm intersect at two points and the distance between their centres is 4 cm. Find the length of the common chord.
If a line intersects two concentric circles (circles with the same centre) with centre O at A, B, C, and D, prove that AB = CD (see given figure).
Three girls Reshma, Salma and Mandip are playing a game by standing on a circle of radius 5m drawn in a park. Reshma throws a ball to Salma, Salma to Mandip, Mandip to Reshma. If the distance between Reshma and Salma and between Salma and Mandip is 6m each, what is the distance between Reshma and Mandip?
Find the length of a chord which is at a distance of 4 cm from the centre of the circle of radius 6 cm.
Two chords AB and CD of lengths 5 cm and 11 cm respectively of a circle are parallel to each other and are opposite side of its center. If the distance between AB and CD is 6 cm. Find the radius of the circle.
If AB is a chord of a circle, P and Q are the two points on the circle different from A and B, then
In a circle of radius 17 cm, two parallel chords are drawn on opposite side of a diameter. The distance between the chords is 23 cm. If the length of one chord is 16 cm, then the length of the other is
Two chords AB and AC of a circle with centre O are on the opposite sides of OA. Then ∠OAB = ∠OAC .
Two equal chords AB and CD of a circle when produced intersect at a point P. Prove that PB = PD.
AB and AC are two chords of a circle of radius r such that AB = 2AC. If p and q are the distances of AB and AC from the centre, prove that 4q2 = p2 + 3r2.
