Two parallel chords of lengths 10 cm and 24 cm are on the same side of the centre of a circle. The distance between the chords is 7 cm. Find the radius of the circle.
13 cm
Explanation
The problem applies the theorem that the perpendicular from the centre of a circle to a chord bisects the chord, forming a right-angled triangle with the radius, half the chord, and the perpendicular distance. By setting up equations for the radius using both chords and the given distance between them, we can solve for the unknown distance and subsequently the radius.
Solution Steps
Step 1: Let the radius of the circle be r. Let the lengths of the two parallel chords be 24 cm and 10 cm. Since the perpendicular from the centre bisects the chord, their half-lengths are 12 cm and 5 cm respectively.
Step 2: Let the distance of the 24 cm chord from the centre be x. Since the chords are on the same side of the centre and the distance between them is 7 cm, the distance of the 10 cm chord from the centre is (x + 7) cm.
Step 3: Using the relation r² = d² + (half chord)², for the 24 cm chord: r² = x² + 12² = x² + 144.
Step 4: For the 10 cm chord: r² = (x + 7)² + 5² = x² + 14x + 74.
Step 5: Equating the two expressions for r²: x² + 144 = x² + 14x + 74. Solving for x, we get 14x = 70, so x = 5 cm.
Step 6: Substituting x = 5 into r² = x² + 144, we get r² = 5² + 144 = 25 + 144 = 169. Therefore, r = 13 cm.