Question

PQ and RS are two parallel chords of a circle, which are on opposite sides of the centre, such that PQ - 6 cm, RS = 8 cm, and distance between PQ and RS is 7 cm Find the radius of the circle ​

Answer 1

Answer:

Firstly, I want to tell you that the distance between them, means that the perpendicular distance between them, i.e., the length of that line, which passes through the centre point of the circle, and is perpendicular to any one chord.

Therefore, if the line passing through the centre point of the circle is perpendicular to one of the chords, then it will also be perpendicular to the other chord, because both the chords are parallel.

So, I start the answer by assuming the length of AO be the variable X, So that the length of OB will be equal to (7-X), because the sum of AO and BO is 7.

So, since the line AB is perpendicular to both the chords, I can clearly see the traingles ∆ AOQ and ∆ BOS are right angled,

It clearly means that (AO)²+(AQ)²= (OQ)²,

and also that, (OB)²+ (BS)²= (OS)².

But at the same time, I also noticed that OQ and OS are both equal to each other, because these are the radii of the same circle.

So, I will now compare both the equations, which I have discussed previously,

(AO)²+ (AQ)²= (OB)²+ (BS)²,

Now, it is a remarkable point that AB is perpendicular to PQ, so, It will bisect PQ,

as the perpendicular from the centre to any chord bisects the chord.

I have drawn F, instead of P, in the figure, but I will mention P here for the point F.

Similarly, BS is also as the half of RS,

So, I again come back to the equation, and put the values of all the line segments.

(PQ/2)²+ (AO)²= (RS/2)²+ (OB)²

SO,

(3)²+ x²= (4)²+ (7-x)²,

= 9+ x²= 16+ (49-14x+x²),

= 9+x²= 16 + 49-14x+x²

I cancel x² from both sides, as they both have the same signs.

= 14x= 16+49-9,

Then, 14x= 16+40,

14x= 56,

then, x= 4 cm.

Now, I take any of the equations, which I have made using the Pythagoras Theorem,

( AO)²+ (AQ)²= (OQ)²,

then, (OQ)²= (x)²+(3)²,

= (4)²+(3)²= (OQ)²,

THEN, (OQ)²= 16+9,

Implies that, (OQ)²= 25,

it gives the value of OQ to be 5 cm.

Since OQ is the radius of the circle, then we can say that, RADIUS OF THIS CIRCLE= 5 CM.

Step-by-step explanation: