Math Problem Statement

The circle below is centered at O. What is the size of angle x? Write a justification for your answer.

Solution

The problem involves finding the angle xx in a circle with its center at OO, where the angle at the center OO is given as 144144^\circ.

Step-by-Step Solution:

  1. Understanding Circle Geometry: The angle xx lies on the circumference of the circle, and it subtends the same arc as the central angle 144144^\circ. By the Angle at the Center Theorem, we know that: The angle at the center is twice the angle at the circumference subtending the same arc.\text{The angle at the center is twice the angle at the circumference subtending the same arc.}

  2. Applying the Theorem: According to the theorem: O=2×x\angle O = 2 \times \angle x Substituting the given value: 144=2×x144^\circ = 2 \times x

  3. Solving for xx: x=1442=72x = \frac{144^\circ}{2} = 72^\circ

Conclusion

The size of angle xx is 7272^\circ.

Justification

This answer follows directly from the rule that the angle at the circumference is half of the angle at the center when both angles subtend the same arc.

Would you like further details or have any questions?

Related Questions:

  1. What is the Angle at the Center Theorem, and how does it apply to circles?
  2. How would you solve for an angle if the given central angle was different?
  3. Can you explain how inscribed angles relate to the diameter of a circle?
  4. What other properties of circles are useful in solving geometric problems?
  5. How can we determine the measure of an exterior angle in a circle?

Tip:

Always remember that the angle subtended by the same arc at the circumference is half the angle subtended at the center of the circle.

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Math Problem Analysis

Mathematical Concepts

Circle Geometry
Angles in Circles

Formulas

Angle at the center = 2 × Angle at the circumference

Theorems

Angle at the Center Theorem

Suitable Grade Level

Grades 8-10