If in two circles, arcs of the same length subtend angles 60° and 75° at the centre, find the ratio of their radii.
The ratio of their radii is 5 : 4.
Explanation
This question is similar to Example 5 in the textbook. The key concept is that for an arc of length l subtending angle θ at the centre of a circle with radius r, the relationship is l = rθ. When two circles have arcs of the same length, we can equate r₁θ₁ = r₂θ₂ and solve for the ratio of radii.
Solution Steps
Step 1: Let r₁ and r₂ be the radii of the two circles. Given that θ₁ = 60° and θ₂ = 75°.
Step 2: Convert angles to radian measure: θ₁ = 60° = 60 × π/180 = π/3 radian θ₂ = 75° = 75 × π/180 = 5π/12 radian
Step 3: Since arcs of the same length subtend these angles, using l = rθ, we have: l = r₁θ₁ = r₂θ₂ Therefore, r₁ × (π/3) = r₂ × (5π/12)
Step 4: Solving for the ratio: r₁/r₂ = (5π/12) ÷ (π/3) = (5π/12) × (3/π) = 5/4
Hence, r₁ : r₂ = 5 : 4.