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 Fig. 2, AB is the diameter of a circle with centre O and AT is a tangent. If ∠AOQ = 58°, find ∠ATQ.
Two tangent segments PA and PB are drawn to a circle with center O such that ∠APB = 120°. Prove that OP = 2AP
O is the centre of a circle of radius 10 cm. P is any point in the circle such that OP = 6 cm. A is the point travelling along the circumference. x is the distance from A to P. what are the least and the greatest values of x in cm? what is the position of the points O, P and A at these values?
In Fig., chords AB and CD of the circle intersect at O. AO = 5 cm, BO = 3 cm and CO = 2.5 cm. Determine the length of DO.
Use the figure given below to fill in the blank:
Diameter = 2 x ________
State, if the following statement is true or false:
The diameters of a circle always pass through the same point in the circle.
The length of tangent from an external point on a circle is always greater than the radius of the circle.
In the following figure, ∠OAB = 30º and ∠OCB = 57º. Find ∠BOC and ∠AOC.
From the figure, identify two points in the interior.
Which is the longest straight line that can be drawn inside a circle?
