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Question
A school wants to conduct tree plantation programme. For this a teacher allotted a circle of radius 6 m ground to nineth standard students for planting sapplings. Four students plant trees at the points A, B, C and D as shown in figure. Here AB = 8 m, CD = 10 m and AB ⊥ CD. If another student places a flower pot at the point P, the intersection of AB and CD, then find the distance from the centre to P.
Solution
OA = OD = 6 m
AB = 8 m ...(chord)
CD = 10 m ...(chord)
In ΔAOM, OM = `sqrt(6^2 - 4^2)` ...(∴ OM bisects the chord and ⊥ to the chord)
= `sqrt(36 - 16)`
= `sqrt20` m
In Δ CON, ON = `sqrt(6^2 - 5^2)`
= `sqrt(36 - 25)`
= `sqrt(11) "m"` ...(ON bisects the chord and ⊥ to the chord)
ONPM is a rectangle with all the angles 90° and with length `sqrt(20)` m, breadth `sqrt(11)` m.
We need to find OP which is the diagonal of the rectangle ONPM.
∴ OP = `sqrt("ON"^2 + "NP"^2)`
= `sqrt((sqrt(11))^2 + (sqrt(20))^2` ...(∴ OM = NP, opposite sides of the rectangle)
= `sqrt(11 + 20)`
= `sqrt(31)`
= 5.56 m
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