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Question
Carom cm. It board has is a very popular game. The board is a square of side length 65 cm. It has circular pockets in each corner. Ansh strikes a disc, kept at position P with a striker. The disc, hits the boundary of the board at R and goes straight to pocket at corner C. It is given that PS = 9 cm, PQ = 35 cm, BR = x, ∠PRQ = α and ∠CRB = θ. |
Based on the above information, answer the following questions:
(i) Using law of reflection i.e. ∠PRT = ∠CRT, prove that θ = α. [1]
(ii) Prove that ΔPQR ~ ΔCBR given that PQ is perpendicular to АВ. [1]
(ii) (a) Find the value of x using similarity of triangles. [2]
OR
(b) If `(Area ΔPQR)/(Area ΔCBR) = (PQ^2)/(CB^2)`, then find the value of x.
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Solution
(i) In the diagram, let line RT be the normal to the boundary AB at point R.
By the law of reflection: ∠PRT = ∠CRT.
The line RT is perpendicular to the side of the carom board, so ∠QRT = ∠BRT = 90°.
Now, ∠PRQ = α = 90° – ∠PRT.
And ∠CRB = θ = 90° – ∠CRT.
Since ∠PRT = ∠CRT, their complements must also be equal:
90° – ∠PRT = 90° – ∠CRT
α = θ
Hence, θ = α.
(ii) In ΔPQR and ΔCBR
∠PQR = ∠CBR = 90° (Given PQ ⊥ AB and the corner of the square carom board is 90°).
∠PRQ = ∠CRB (Proved in part (i) as α = θ).
Therefore, by AA Similarity Criterion: Two triangles are similar if two of their corresponding angles are equal.
ΔPQR ∼ ΔCBR
Hence proved.
(iii) (a) Side of square board = 65 cm. Thus, CB = 65.
Given PQ = 35.
From diagram, S is on AD and PQ is perpendicular to AB. PS = 9 cm represents the distance of Q from corner A.
So, AQ = 9.
Since AB = 65, the length QB = 65 – 9 = 56.
We are given BR = x. Since R is on the segment QB, QR = QB – BR = 56 – x.
From ΔPQR ∼ ΔCBR
`(PQ)/(CB) = (QR)/(BR)`
`35/65 = (56 - x)/x`
`7/13 = (56 - x)/x`
7x = 13(56 – x)
7x = 728 – 13x
20x = 728
`x = 728/20`
x = 36.4 cm
The value of x is 36.4 cm.
(iii) (b) ΔPQR ∼ ΔCBR
The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.
`(PQ)/(CB) = (QR)/(BR)`
`35/65 = (56 - x)/x`
⇒ x = 36.4 cm
The value of x is 36.4 cm.
