Calculate arc lengths, angles and areas of sectors of circles (2)
This is the second of two weekly sets of tasks looking at arcs and sectors. In this set we focus on the angles subtended by and the lengths of arcs.
MONDAY: This is a straightforward task but it provides a nice visual exercise of considering what might happen if we try to wrap the green arc over the red arc (or open up the red arc to fit under the green arc). Might they fit exactly? The task hinges on the observation that the radius (and hence the circumference) of the larger circle is twice the radius (and the circumference) of the smaller circle.
The green arc would fit exactly over the red arc: half the circumference of the smaller circle is equal to a quarter of the circumference of the larger circle as its circumference is twice as large!
TUESDAY: Monday's task can be seen as a special case of this more general task. It is rather wonderful that the relation between the arc lengths still holds!
Angle CAD is the angle (g˚) subtended by the green arc at the centre of its circle. It is also the angle subtended by the red arc at the circumference (c units) of its circle, and so is half the size of the angle subtended at the centre.
We can write the length of the green arc as 2c × g/360, and the length of the red arc as c × 2g/360; the expressions are equivalent.
The task can be made more general still (below). The relation still holds.
WEDNESDAY: Here we have two identical circles, but the angle subtended by the green arc at its circle's centre is again twice that of the red arc.
This time the green arc is twice the length of the red arc. The relation still holds in this more general case (below).
THURSDAY: Another variant, again involving identical circles.
The angle subtended by the red arc at the centre of its circle is also the angle between the chord bounding the green arc and a tangent to its circle. So this angle is half the angle subtended by the green arc at the centre of its circle (on the basis of the Angle between tangent and chord theorem). Therefore the green arc is twice the length of the red arc.
FRIDAY: This task hinges on the fact that the circumference of the largest semi-circle is equal to the sum of the circumferences of the semi-circles that the largest semi-circle contains. [We explore this phenomenon in Task 05E of MultipliXing.]
In the situation shown in the diagram below, the sum of the lengths of the red arcs is equal to the length of the green arc. We can show this algebraically:
Total length of red arcs = 𝛑r₁ × 2a/360 + 𝛑r₂ × 2a/360 = 𝛑(r₁ + r₂) × 2a/360 = 𝛑R × 2a/360 = length of green arc.
However, in the original task, the angle EBC is greater than a˚ and so the total length of the red arcs is greater than the length of the green arc.
EXTRA: In this task we don't really need three diagrams, but students might find it illuminating to see that the lengths of the two green arcs are fixed even though they occupy different parts of the circle in the different diagrams.
