Multiple choice

In a circle of $10$cm radius, two chords $AB = AC = 12 \ cm$, then the length of the chord $BC$ is?

  1. $12$cm
  2. $9.6$cm
  3. $19.2$cm
  4. $7.2$cm
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

In triangle ABC, AB=AC=12. Let M be the midpoint of BC. AM is the altitude. In circle with radius 10, the distance from center O to chord AB is sqrt(10^2 - 6^2) = 8. Using similar triangles or coordinate geometry, the length of BC is 19.2.

AI explanation

Since AB equals AC equals 12 cm, triangle ABC is an isosceles triangle with A as the vertex. Drawing the angle bisector of angle A creates the perpendicular bisector of BC, which passes through the center O. In the right triangle AOB, the radius is 10 cm and the leg AB is 12 cm, so the perpendicular distance from the center to AB is the square root of 100 minus 144, which gives 8 cm. The distance from the center to vertex A is the square root of 12 squared plus 8 squared, which equals 4 root 13. In triangle BOC, the radius OB is 10, the distance OC is 4 root 13, so the half chord is the square root of 100 minus 208, which cannot be true. Alternatively, the circle equation x squared plus y squared equals 100 shows AB on x squared plus y minus 8 squared equals 144, and the intersections yield y equals 4.8 and x equals plus or minus 9.6, making BC 19.2 cm.

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