Circle

Exercise 10.6

Part 1

Question 1: Prove that the line of centres of two intersecting circles subtends equal angles at the two points of intersection.

Answer: The following figure shows two circles with centres O and O' and the circles are intersecting at points P and Q.

To Prove: ∠OPO' = ∠OQO'

In ΔOPO' and ΔOQO'

OP = OQ (Radii of same circle)

O'P = O'Q (Radii of same circle)

OO' = OO' (common side)

So, ΔOPO' ≅ ΔOQO' (SSS Theorem)

So, ∠OPO' = ∠OQO' Proved

Question 2: Two chords AB and CD of lengths 5 cm and 11 cm respectively of a circle are parallel to each other and are on opposite sides of its centre. If the distance between AB and CD is 6 cm, find the radius of the circle.

Answer: The following figure shows a circle with center O, in which AB and CD are parallel chords. AB = 5 cm, CD = 11 cm and perpendicular distance between AB and CD = 6 cm

Let us draw PQ the perpendicular distance between AB and CD so that this line passes through centre O.

Now, join A and C to O

In Δ APO:

AP = `(AB)/2=2.5` cm (Line from center is perpendicular bisector of chord)

`OA=r`

`OP=6-x` (If `OQ=x`)

Applying Pythagoras theorem, we get

OA² - OP² = AP²

Or, `r^2-(6-x)^2=2.5^2`

Or, `r^2-(36+x^2-12x)=6.25`

Or, `r^2-36-x^2+12x=6.25`

Or, `r^2=6.25+36+x^2-12x`………………(1)

In ΔCQO:

`CQ=(CD)/2=5.5` cm (Line from center is perpendicular bisector of chord)

`OQ =x`

Applying Pythagoras theorem, we get

OC² = OQ² + CQ²

Or, `r^2=x^2+5.5^2`

Or, `r^2=x^2+30.25` ……………..(2)

From equations (1) and (2)

`x^2+30.25=x^2-12x+42.25`

Or, `-12x+42.25=30.25`

Or, `-12x=30.25-42.25=-12`

Or, `x=1`

Substituting the value of x in equation (1) or (2), we can calculate the value of r as follows:

`r^2=1^2+30.25=31.25`

Or, `r=sqrt(31.25)=5.57` (approx)

Question 3: The lengths of two prallel chords of a circle are 6 cm and 8 cm. If the smaller chord is at distance 4 cm from the centre, what is the distance of the other chord from the centre?

Answer: The question does not specify whether the chords are on the same side of center or on opposite sides. We will solve by assuming both conditions.

This figure shows a circle with centre O. Two parallel chords AB and CD are on the same side of center and AB is 4 cm from the centre.

In ΔAPO

AO² = AP² + PQ²

Or, `r^2=3^2+4^2`

(AP = `6/2=3` cm because PO is perpendicular bisector of AB)

Or, `r^2=9+16=25`

Or, `r=5` cm

In ΔCQO

OQ² = CO² - CQ²

Or, `OQ^2=r^2-4^2`

(CQ = `8/2=4` because OQ is perpendicualr bisector of CD)

Or, `OQ^2=5^2-4^2=25016-9`

Or, OQ = 3 cm

Let us now assume that the chords are on opposite sides of the centre, as shown in this figure.

In ΔAPO:

AP = 3 cm and OP = 4 cm

AO² = 4² + 3² = 16 + 9 = 25

Or, AO = 5 cm

In ΔCQO:

CQ = 4 cm and CO = 5 cm (radius)

OQ² = CO² - CQ²

= 5² - 4² = 25 – 16 = 9

Or, OQ = 3 cm