Q) In a park, four poles are standing at positions A, B, C and D around the circular fountain such that the cloth joining the poles AB, BC, CD and DA touches the circular fountain at P, Q, R and S respectively as shown in the figure.

Based on the above information, answer the following questions :
(i) If O is the centre of the circular fountain, then ∠ OSA =
(ii) If AB = AD, then write the name of the figure ABCD.
(iii) (a) If DR = 7 cm and AD = 11 cm, then find the length of AP.
OR (iii) (b) If O is the centre of the circular fountain with ∠ QCR = 60°, then find the measure of ∠ QOR.

Ans: (i) Value of ∠ OSA:

Let’s connect OS in the diagram:

We know that the radius is always perpendicular to the tangent at the point of contact,

∴  ∠ OSA = 90°

Hence value of ∠ OSA is 90°

(ii) Name of Quadrilateral ABCD:

In the given diagram, we are given AB = AD,

We know that AB = AP + PB and AD = AS + SD

By substituting these values, we get:

∵ AB = AD

∴ AP + PB = AS + SD

Since AS and AP are the tangents on a circle from the same point A, hence, AS = AP

∴ AP + PB = AP + SD

∴ PB = SD ………… (i)

Next, We can see that SD and RD are the tangents on a circle from the same point D, hence, SD = RD

and PB and QB are the tangents on a circle from the same point B, hence, PB = QB

By substituting these values in equation (i), we get:

∵ PB = SD

∴ QB = RD

We add RC on both sides, we get:

QB + RC = RD + RC

Since, RC and QC are the tangents on a circle from the same point C, hence, RC = QC

∴ QB + QC = RD + RC

∴ BC = CD 

Now, AB and AD are adjacent sides and equal (by given condition).

Similarly, BC and CD are adjacent sides (solved above)

Therefore, given shape is a kite.

[Note: the given shape could also be Square or Rhombus. But for that, we need to check whether all 4 sides are equal or not, whether the angles of the quadrilateral are 90 or not, whether diagonals bisecting or not, etc. But we do not have sufficient information in the question regarding that and hence with the given information, it can not be said that the given shape is a Square or a Rhombus.]

(iii) (a) Length of AP:  Let’s check the diagram again:

Here, DR and DS are the tangents to the circle from an external point D,

Therefore DR = DS (tangents to a circle from a point are equal)

Given that DR = 7 cm, ∴ DS = 7 cm

Next, we are given AD = 11 cm

∵ AD = AS + DS

∴ AS = AD – DS = 11 – 7 = 4 cm

Next, we have AP and AS tangents to the circle from an external point A, therefore AP = AS

Since, we just calculated, AS = 4 cm, ∴ AP = 4 cm

Therefore, length of AP is 4 cm.

(iii) (b) Value of ∠ QOR:

Let’s connect OQ and OR:

OQCR forms a quadrilateral.

Next, as stated above, the radius is always perpendicular to the tangent at the point of

contact, ∴  ∠ ORC = 90° and ∠ OQC = 90°

Next, we know that the sum of all 4 angles in a quadrilateral is 360°

∴ ∠ OQC + ∠ QCR +∠ ORC + ∠ QOR = 360°

We calculated above values of ∠ ORC and ∠ OQC as 90°

and we are given that ∠ QCR = 60°

By substituting values of these angles, we get:

90° + 60° + 90° + ∠ QOR = 360°

∴ ∠ QOR = 360° – 240° = 120°

Therefore, the value of ∠ QOR is 120°

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