Multiple choice

In a circle of radius 10 cm, with centre O, PQ and PR are two chords each of length 12 cm. PO intersects chord QR at the point S. The length of OS is

  1. 2.8 cm

  2. 2.5 cm

  3. 3.2 cm

  4. 3 cm
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A Correct answer
Explanation

In a circle, the perpendicular from the center to a chord bisects the chord. QR is a chord. Since PQ=PR=12, triangle PQR is isosceles. PO is the perpendicular bisector of QR. Using Pythagoras in triangle OMQ (where M is midpoint of PQ), find the distance.

AI explanation

Draw the perpendicular from O to PQ, which bisects the 12 cm chord PQ into two halves of 6 cm each; applying the Pythagorean theorem gives the distance from O to PQ as the square root of (10 squared minus 6 squared), which is 8 cm. Since chords PQ and PR are equal in length, the line PO bisects the angle QPR and the arc QR, making PO the perpendicular bisector of QR. In the right triangle formed by O, the midpoint of PQ, and P, we find that OP is 10 cm, the distance to the chord is 8 cm, and half of PQ is 6 cm, meaning OS bisects QR; using geometric properties of the intersecting chords and similar triangles, OS calculates to 2.8 cm.

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