Which one of the following is <strong>CORRECT</strong> in respect of angle \theta?

Consider the following for the next three (03) items that follow:<br>Two circles with centres at O_{1} and O_{2} touching each other are placed inside a rectangle of sides 9 cm and 8 cm as shown in the figure given below.

  1. A. 0^{\circ} \lt \theta \lt 30^{\circ}
  2. B. 30^{\circ} \lt \theta \lt 45^{\circ} ✓
  3. C. 45^{\circ} \lt \theta \lt 60^{\circ}
  4. D. 60^{\circ} \lt \theta \lt 90^{\circ}

Correct Answer: B. 30^{\circ} \lt \theta \lt 45^{\circ}

Explanation

Using the centers' coordinates, the vertical difference \Delta y = 2.5 and the horizontal difference \Delta x = 3.5. Thus, \tan \theta = \frac{2.5}{3.5} = \frac{5}{7} \approx 0.71. Since \tan 30^{\circ} \approx 0.577 and \tan 45^{\circ} = 1, the angle \theta lies strictly between 30^{\circ} and 45^{\circ}.

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