Math Problem Statement

O is the centre of this circle. Calculate the size of angle x. Show all of your working and state any theorems that you use.

Solution

To calculate the angle xx, we will use key circle theorems:

  1. Theorem Used: The angle at the center of a circle is twice the angle at the circumference when subtended by the same arc.

Step-by-Step Solution:

  • Step 1: Identify which angles are subtended by the same arc.

    • Angle NKJ=128\angle NKJ = 128^\circ (at the circumference) and angle xx (at the center OO) are subtended by arc NJNJ.
  • Step 2: Apply the theorem:

    The angle at the center is twice the angle at the circumference.

x=2×NKJx = 2 \times \angle NKJ

Substitute the value of NKJ\angle NKJ:

x=2×128x = 2 \times 128^\circ

x=256x = 256^\circ


Final Answer:

The size of angle xx is 256°.


Do you want further clarification or additional explanations? Let me know!

Here are 5 related questions to explore:

  1. How do you prove that the angle at the center is twice the angle at the circumference?
  2. What is the sum of interior angles in a quadrilateral inscribed in a circle?
  3. Can you find other relationships between angles in the circle diagram?
  4. What happens if the quadrilateral is cyclic? What properties arise?
  5. How can the circle theorems be applied to find missing angles in other geometric figures?

Tip: Remember that the angle at the center is always twice the angle at the circumference when subtended by the same arc.

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Math Problem Analysis

Mathematical Concepts

Circle Geometry
Angles at the Centre and Circumference

Formulas

Angle at the center = 2 × Angle at the circumference

Theorems

The angle at the center of a circle is twice the angle at the circumference subtended by the same arc

Suitable Grade Level

Grades 9-10