In a circle with centre O, the central angle AOB is 60°. If the radius of the circle is 12 cm, what is the length of the chord AB?
Step 1: In the circle with centre O, OA and OB are the radii, so OA = OB = 12 cm. This makes triangle OAB an isosceles triangle.
Step 2: The central angle subtended by the chord AB at the centre is ∠AOB = 60°. Since OA = OB, the angles opposite to the equal sides are equal. Therefore, ∠OAB = ∠OBA.
Step 3: Using the angle sum property: ∠OAB + ∠OBA + ∠AOB = 180°. So, 2(∠OAB) + 60° = 180°, which gives 2(∠OAB) = 120°, meaning ∠OAB = 60°.
Step 4: Since all angles of triangle OAB are 60°, it is an equilateral triangle. Thus, all its sides are equal. Length of chord AB = OA = OB = 12 cm.
Explanation
The student identifies that triangle OAB is isosceles because OA and OB are radii of the same circle. Given the central angle is 60°, the base angles opposite the equal sides must also be 60° each, making it an equilateral triangle. Hence, the chord length equals the radius.
Solution Steps
Step 1: Identify that OA = OB = 12 cm (radii), making △OAB an isosceles triangle.
Step 2: Given ∠AOB = 60°, and since ∠OAB = ∠OBA (angles opposite equal sides), we have 2(∠OAB) + 60° = 180°.
Step 3: Solving gives ∠OAB = 60° and ∠OBA = 60°, so △OAB is an equilateral triangle.
Step 4: Conclude that AB = OA = OB = 12 cm.