Question

Carom board 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 $θ = α$ .

(ii) Prove that $△ PQR \sim △ CBR$ given that PQ is perpendicular to AB.

(iii) (a) Find the value of $x$ using similarity of triangles.

(b) If $\frac{Area △ PQR}{Area △ CBR} = \frac{P Q ^{2}}{C B ^{2}}$ , then find the value of $x$ .

Approach

Carom board problem: disc at P(9, 0) hits boundary at R and goes to pocket C(0, 65). Given $PS = 9$ cm, $PQ = 35$ cm, $BR = x$ , $∠ PRQ = α$ , $∠ CRB = θ$ . (i) Using law of reflection (angle of incidence = angle of reflection), $∠ PRT = ∠ CRT$ , prove $θ = α$ . (ii) Since $∠ PRQ = ∠ CRB$ and both have right angles at Q and B, $△ PQR \sim △ CBR$ . (iii)(a) From similarity, $\frac{PQ}{CB} = \frac{QR}{BR}$ . Solving gives $x = 36.4$ cm. OR (iii)(b) Using area ratio, $\frac{Ar ( △ PQR )}{Ar ( △ CBR )} = \frac{P Q ^{2}}{C B ^{2}}$ , solve for $x = 36.4$ cm.

Step-by-step working

1
(i) Draw tangent TR at R. By law of reflection, $∠ PRT = ∠ CRT$ . Since $TR ⊥ A B$ , $α + ∠ PRT = θ + ∠ CRT$ . Hence $α = θ$ .
1 mark
2
(ii) Since $θ = α$ , $∠ PRQ = ∠ CRB$ . Also $∠ PQR = ∠ CBR = 90°$ . By AA similarity, $△ PQR \sim △ CBR$ .
1 mark
3
(iii)(a) From similarity: $\frac{PQ}{CB} = \frac{QR}{BR}$ . $PQ = 35$ cm, $CB = 65$ cm, $QR = 65 - 9 - x = 56 - x$ . $\frac{35}{65} = \frac{56 - x}{x}$ .
0.5 marks
4
Cross-multiply: $35 x = 65 (56 - x) \Rightarrow 35 x = 3640 - 65 x \Rightarrow 100 x = 3640 \Rightarrow x = 36.4$ cm.
1 mark
5
OR (iii)(b) $\frac{Ar ( △ PQR )}{Ar ( △ CBR )} = \frac{P Q ^{2}}{C B ^{2}}$ . $\frac{\frac{1}{2} \times 35 \times ( 56 - x )}{\frac{1}{2} \times 65 \times x} = \frac{3 5 ^{2}}{6 5 ^{2}}$ .
1 mark
6
Simplify: $\frac{56 - x}{x} = \frac{35}{65} \Rightarrow 65 (56 - x) = 35 x \Rightarrow x = 36.4$ cm.
0.5 marks

Concepts used

similar triangleslaw of reflectionprovefind value

Approach

Step-by-step working

Concepts used

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