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МатематикаВопрос:
In the given figure, O is the center of the circle. If m∠MOK = 78°, find the measure of ∠MNK.
Ответ:
Insight:
Method: The angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle.
Step-by-step solution:
- Step 1: Identify the arc subtending the angles. Arc MK subtends ∠MOK at the center and ∠MNK at the circumference.
- Step 2: Apply the theorem relating the central angle and the inscribed angle subtended by the same arc. The theorem states that the angle at the center is twice the angle at the circumference. Therefore, $$m∠MOK = 2 imes m∠MNK$$.
- Step 3: Substitute the given value of ∠MOK and solve for ∠MNK. Given $$m∠MOK = 78°$$. So, $$78° = 2 imes m∠MNK$$.
- Step 4: Calculate the measure of ∠MNK. $$m∠MNK = 78° / 2 = 39°$$.
Answer: 39°
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