Question

In the figure, ‘O’ is the center of the circle and OM, ON are the perpendiculars from the center to the chords, PQ and RS. If OM = ON and PQ = 6 cm. Find RS.

Answer 1

Hint: In order to solve this question, we will consider the triangles OPM and ORN and by using congruence, we will find the length of RN and we will use the rule that perpendicular from the center bisects the chord. By using these concepts we will find the solution of this question.
Complete step-by-step answer:
In this question, we have been asked to find the length of RS when we have been given that ON and OM are perpendiculars on the chords PQ and RS and it is given that OM = ON and PQ = 6 cm. To solve this question, we will first do the construction in the given figure by joining P and O and R and O. So, we get the figure as,

Here, we have OP and OR as the radius of the circle, PQ and RS as the chords of the circle. Now, we know that perpendicular on the chord from the center bisects the chord. So, we can say that,
$PM=\dfrac{1}{2}PQ.........\left( i \right)$ and $RN=\dfrac{1}{2}RS.........\left( ii \right)$
Now, let us consider triangle OPM and triangle ORS. So, we know that, $\angle OMP=\angle ONR={{90}^{\circ }}$ which is given to us in the question. Also, we have been given that OM = ON. We know that OP and OR are the radius of the circle. So, OP = OR. From the above three equations, we can say that, $\Delta OPM\cong \Delta ORN$ by RHS axiom congruence. Therefore, we can say that PM = RN.
Now we will take the values of PM and RN from the equations (i) and (ii) and substitute it in the equation, PM = RN. So, we get,
$\dfrac{1}{2}PQ=\dfrac{1}{2}RS$
PQ = RS
Now, we know that PQ = 6 cm. So, we get,
RS = 6 cm
Hence, the value of RS is 6 cm.

Note: While solving this question, we can also use a property that “chords of a circle, which are at equal distance from the center are equal in length”. So, we can directly say that PQ = RS, which implies RS = 6 cm.