ACTIVITY 2 2.1 Draw two circles with centre $$ O $$, and radii of 3 cm and 5 cm respectively 2.1.1 Mark any two pòints $$ P $$ and $$ Q $$ on the circumference of circle to create a minor and major arc in each of the circles. 2.1.2 Draw the radii, P to O and Q to O in both circles to produce angle $$ \hat{\mathrm{O}}_{1} $$ and a reflex angle $$ \hat{\mathrm{O}}_{2} $$ 2.1.3 Mark point R on the circumference of each circle, on the opposite segment to $$ P $$ and $$ Q $$. 2.1.4 Join P to R and Q to R in both circles. 2.1.5 Measure the sizes of angles PRQ and $$ \hat{\mathrm{O}}_{1} $$ in both circles 2.1.6 Carefully study your answers in 2.1 .5 and make a conclusion.

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Below is one way you (or your students) might work through Activity 2 and explain your conclusions.

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Step 1. Draw the Two Circles

• Draw circle A with centre O and a radius of 3 cm.
• Draw circle B with the same centre O (or a different centre if you wish, but for clarity they are often drawn separately) and a radius of 5 cm.

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Step 2. Mark Two Points on Each Circumference

• On circle A, choose any two distinct points on the circumference and label them P and Q. Choose these so that one arc (the shorter one) is a minor arc and the other (the larger one) is the major arc.
• Do the same for circle B. (The positions of the points may be the same relative to the centre, or different, as long as you have a clear minor and a major arc.)

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Step 3. Draw the Radii and Identify the Angles

• In each circle, join O to P and O to Q. In the first circle, label the angle between OP and OQ as ∠O1.
• In the second circle, if you draw the two radii you can see that if you “travel the long way” around the circle the angle you get is a reflex angle; label that as ∠O2.

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Step 4. Locate Point R

• In each circle, choose a point R on the circumference that lies in the segment opposite to the segment containing P and Q. In other words, R should be on the arc that is “cut off” or separated from the arc containing P and Q by the chord joining P and Q.

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Step 5. Draw the Chords

• In each circle, join P to R and Q to R.

You now have a triangle P R Q inscribed in each circle along with the two radii OP and OQ in each.

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Step 6. Measure the Angles

• Using your protractor, measure the angle at point R in triangle PRQ (that is, ∠PRQ) in both circles.
• Also, measure the central angle ∠O1 (the one between OP and OQ corresponding to the smaller, or minor, arc) in the first circle. (If desired, measure the corresponding central angle in the second circle as well—or use the reflex angle carefully keeping in mind its “wrap‐around” nature.)

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Step 7. Analyze and Draw Your Conclusion

When you compare the measured size of the inscribed angle ∠PRQ with the central angle (or the corresponding angle that subtends the same arc), you should notice the following relationship:

The measure of an inscribed angle (∠PRQ) is exactly half of the measure of its corresponding central angle (∠O1).

This is precisely the Inscribed Angle Theorem. Note that this relationship holds true regardless of the size (radius) of the circle. In other words, both the 3‑cm and 5‑cm circles show that if an arc (or chord subtending an arc) gives rise to an inscribed angle, that inscribed angle is always half the measure of the central angle subtending the same arc.

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Conclusion

When you studied your answers in step 2.1.5, you observed that in both circles the angle formed at a point on the circumference (∠PRQ) measured half of the corresponding central angle (∠O1). In summary, the construction and measurements confirm that in any circle (no matter its size) the inscribed angle is always half the central angle that subtends the same chord.

This finding is an illustration of the Inscribed Angle Theorem.

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Additional Notes

• If you measured a reflex angle in one of the cases (for example, ∠O2) you must be careful to recognize that the inscribed angle theorem applies to the “central” angle subtending the same arc (usually the smaller one, even if the point R is chosen in the “opposite” segment).
• It is useful to redo the experiment with different choices for points P, Q, and R to reinforce the consistency of this relationship.

This completes Activity 2 and demonstrates the reliability of the inscribed angle theorem in circles of different sizes.
When you draw two circles with centers O and radii 3 cm and 5 cm, mark points P and Q on each to create minor and major arcs, draw radii to form angles ∠O₁ and reflex ∠O₂, and then draw chords PR and QR, you'll find that the inscribed angle ∠PRQ is always half the measure of the central angle ∠O₁ in both circles. This confirms the Inscribed Angle Theorem, which states that in any circle, the inscribed angle is half the central angle subtending the same arc, regardless of the circle's size.

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