Question

q and R are the centres of two congruent circles intersecting each other at point c and d the line joining the circles intersect the circles in points A and B such that a and b do not lie between Q and R if CD is equal to 6 cm and AD is equal to 12 cm determine the radius of either circles and the distance between the centres of the two circles​

Answer 1

Q and R are the centres of two congruent circles intersecting each other at point c and d the line joining the circles intersect the circles in points A and B such that a and b do not lie between Q and R.

So, we have,

Radius of the circle = r = AQ = QC = QD = RC = RB = RD

QCRD is a rhombus ( ∵ QC = QD = RC = RD )

Since diagonals of a rhombus are ⊥ar bisectors, so we have,

CD ⊥ QR

⇒ QR = RP  and CP = PD

CD = 6 cm ⇒ CP = CD/2 = 3 cm

AB = 12 cm

QP = QR/2

AB = AQ + QR + QB

QR = AB - AQ - RB

⇒ QR = 12 - r - r

⇒ QR = 2( 6 - r )

So, QP = 2( 6 - r ) / 2 = (6 - r) cm

Now consider, In Δ QPC,

QC² = CP² + OP²

= 3² + (6 - r)²

In Δ QPC,

QC² = CP² + QP²

r² = 3² + (6 - r)²

r² = 9 + 36 + r² - 12r

12r = 45

r = 3.75 cm.

Since the circles are concentric, r1 = r2 = r = 3.75 cm