Advertisements
Advertisements
Question
If a line segment joining mid-points of two chords of a circle passes through the centre of the circle, prove that the two chords are parallel.
Advertisements
Solution
Given: AB and CD are two chords of a circle whose centre is O and PQ is a diameter bisecting the chord AB and CD at L and M, respectively and the diameter PQ passes through the centre O of the circle.
To prove: AB || CD
Proof: Since, L is the mid-point of AB.
∴ OL ⊥ AB ...[Since, the line joining the centre of a circle to the mid-point of a chord is perpendicular to the chord]
⇒ ∠ALO = 90° ...(i)
Similarly, OM ⊥ CD
∴ ∠OMD = 90° ...(ii)
From equations (i) and (ii),
∠ALO = ∠OMD = 90°
But, these are alternating angles.
So, AB || CD
Hence proved.
APPEARS IN
RELATED QUESTIONS
In the given figure, PQ is a chord of length 8cm of a circle of radius 5cm. The tangents at P and Q intersect at a point T. Find the length TP
Two tangents TP and TQ are drawn to a circle with centre O from an external point T. Prove that ∠PTQ = 2∠OPQ.
Find the length of the tangent drawn from a point whose distance from the centre of a circle is 25 cm. Given that the radius of the circle is 7 cm.
If the tangent at point P to the circle with center O cuts a line through O at Q such that PQ= 24cm and OQ = 25 cm. Find the radius of circle
From a point P, two tangents PA and PB are drawn to a circle with center O. If OP = diameter of the circle shows that ΔAPB is equilateral.
In fig. a circle touches all the four sides of quadrilateral ABCD with AB = 6cm, BC = 7cm, CD = 4cm. Find AD.
In fig. there are two concentric circles with Centre O of radii 5cm and 3cm. From an
external point P, tangents PA and PB are drawn to these circles if AP = 12cm, find the
tangent length of BP.
In Figure 3, a circle touches all the four sides of a quadrilateral ABCD whose sides are AB = 6 cm, BC = 9 cm and CD = 8 cm. Find the length of the side AD.
If A, B, C and D are four points such that ∠BAC = 45° and ∠BDC = 45°, then A, B, C, D are concyclic.
Say true or false:
The centre of a circle is always in its interior.
