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Two circles of radii 15 cm and 12 cm intersect each other, and the length of their common chord is 18 cm. What is the distance (in cm) between their centers?

Two circles of radii 15 cm and 12 cm intersect each other, and the length of their common chord is 18 cm. What is the distance (in cm) between their centers? Correct Answer <span class="math-tex">\(12 + 3\sqrt 7\)</span>

Let O and O’ be the center of the circle and PQ be the length of the common chord.

OP = 15, O’P = 12

PM = ½ PQ (as perpendicular from the center of the circle to a chord bisects the chord)

PM = ½ 18

PM = 9

By applying Pythagoras theorem in PMO:

OM2 = OP2 – PM2

= 152 - 92

= 225 – 81

= 144

OM = √144 = 12

By applying Pythagoras theorem in PMO’:

O’M2 = O’P2 – PM2

= 122 - 92

= 144 – 81

= 63

O’M = √63

O’M = 3√7

OO’ = OM + O’M

= 12 + 3√7

∴ The distance between their centers = 12 + 3√7

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