Instructions.
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Use the slider
Radius 1 (r₁)to change the size of the first circle. -
Use the slider
Radius 2 (r₂)to change the size of the second circle. -
Use the slider
Distance (P₁P₂)to move the centers closer together or farther apart. -
Observe when the circles:
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Do not touch.
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Touch at one point.
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Overlap to form a shaded region.
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When the circles overlap, observe the shaded region (the lens) and the angles shown.
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Examine the calculation panel to see how the shared area is determined.
Use the interactive to explore and answer the following:
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What happens when the distance between the centers is greater than the sum of the radii (\(d > r_1 + r_2\))? What do you observe about the shaded region?
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What happens when the distance is less than the difference of the radii (\(d < |r_1 - r_2|\))? How are the circles positioned?
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Adjust the sliders so the circles just touch at one point. What is the area of overlap in this case?
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When the circles overlap, how does increasing the distance \(d\) affect the size of the shared region?
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Keep the distance fixed and increase one of the radii. How does this change the overlap? Which circle contributes more to the shared region?
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Observe the angles \(\theta\) and \(\alpha\text{.}\) How do these angles change as the overlap increases or decreases?
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The shared region is made up of two segments. How is each segment related to its corresponding circle?
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Based on your observations, explain why the total shared area can be found by adding two segment areas together.
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Can you describe the conditions needed for two circles to intersect? How can these conditions be written using \(r_1\text{,}\) \(r_2\text{,}\) and \(d\text{?}\)