Question

In two concentric circles, chord AB of the outer circles cuts the inner circle at C and D. Prove that AC=BD.

Answer 1

Hint: In order to prove this question we will first draw a perpendicular on a line and proceed further by using the property of the circle as the perpendicular drawn from the center of a circle to a chord bisects the chord of the circle.

Complete step-by-step answer:

Let a line intersects two concentric circles with center O at A, B, C and D .

To Prove:
AB= CD

Construction – Draw OM perpendicular from O on a line.

Proof:

We know that the perpendicular drawn from the center of a circle to the chord bisects the chord.

Here, AD is a chord of a larger circle.
$\text{OM}\mathrm{\perp }\text{AD}$ is drawn from O.
OM bisects AD as $\text{OM}\mathrm{\perp }\text{AD}$
$\text{AM}=\text{ MD}.........\text{(1)}$

Here, BC is the chord of the smaller circle.
OM bisects BC as $\text{OM}\mathrm{\perp }\text{ BC}$ .
$\text{BM}=\text{MC}..........\text{(2)}$

From (1) and (2),

On subtracting equation (1) and from (2)
$$\begin{gathered} \text{AM - BM}=\text{MD - MC} \\ \text{AB}=\text{CD} \end{gathered}$$

Hence, $\text{AB}=\text{CD}$

Note: In order to solve this question, we use the property of the circles. So remember all the properties of the circles. Also remember when a line bisects the other line perpendicularly; it divides the line into two equal parts. Also be familiar with the terms like chord, secant, tangent. A secant is simply a line that intersects two points of the circle.