Case Study - 1
36. 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.
(tangents, circle, quadrilateral, angle, inscribed circle)
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Step-by-Step Solution
Step 1: Part (i): Angle between radius and tangent
According to the tangent-radius theorem, the radius drawn to the point of tangency is perpendicular to the tangent. Here, OS is the radius and DA is the tangent at point S. Therefore, the angle formed between them, ∠OSA, is 90∘.
Step 2: Part (ii): Identifying the quadrilateral
A quadrilateral is a kite if two pairs of equal-length sides are adjacent to each other. We are given that AB=AD. Also, from the property that tangents from an external point to a circle are equal in length, we have AP=AS, BP=BQ, CQ=CR, and DR=DS. If AB=AD, then the quadrilateral ABCD is a kite.
Step 3: Part (iii) (a): Finding length of AP
We know that tangents from an external point to a circle are equal in length. So, DS=DR=7 cm. The side AD is composed of AS and DS. Thus, AS=AD−DS=11 cm−7 cm=4 cm. Since AP and AS are tangents from point A, AP=AS. Therefore, AP=4 cm.
Step 4: Part (iii) (b): Finding angle QOR
In quadrilateral OQCR, ∠OQC and ∠ORC are 90∘ because the radius is perpendicular to the tangent at the point of contact. The sum of angles in a quadrilateral is 360∘. So, ∠QOR+∠OQC+∠QCR+∠ORC=360∘. Substituting the known values, ∠QOR+90∘+60∘+90∘=360∘. This simplifies to ∠QOR+240∘=360∘, which gives ∠QOR=120∘.