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  1. Problems
  2. /C and D are two points on the major arc AB of a circle with centre O…
MediumMathematicsHigh School (9–10)GeneralGeometry › CirclesNumericalSAT

C and D are two points on the major arc AB of a circle with centre O…

StemPy LabContributed by StemPy Lab

Try it on paper first. A hint, the key idea and the worked solution are here when you need them.

CCC and DDD are two points on the major arc ABABAB of a circle with centre OOO. ∠ACB=(3x+6)∘\angle ACB = (3x + 6)^\circ∠ACB=(3x+6)∘ and ∠ADB=(5x−18)∘\angle ADB = (5x - 18)^\circ∠ADB=(5x−18)∘. Find xxx and ∠AOB\angle AOB∠AOB.

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Work it out on paper, then type your number. The answer itself stays hidden.

Need a reminder? The key ideaShowHide

The angle at the centre is double the angle at any point on the remaining part of the circle. Angles in the same segment are equal. The angle in a semicircle is 90∘90^\circ90∘, and the hypotenuse of a right triangle is a diameter of the circle through its corners.

The worked solution

Have you tried it?

Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.

Step 1. The equal angles. CCC and DDD lie in the same segment, so ∠ACB=∠ADB\angle ACB = \angle ADB∠ACB=∠ADB.

Step 2. Solve. 3x+6=5x−183x + 6 = 5x - 183x+6=5x−18 gives 24=2x24 = 2x24=2x, so x=12x = 12x=12.

Step 3. The rim angle. ∠ACB=3×12+6=42∘\angle ACB = 3 \times 12 + 6 = 42^\circ∠ACB=3×12+6=42∘.

Step 4. The centre. ∠AOB=2×42∘=84∘\angle AOB = 2 \times 42^\circ = 84^\circ∠AOB=2×42∘=84∘.

Answer: x=12x = 12x=12 and ∠AOB=84∘\angle AOB = 84^\circ∠AOB=84∘.

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