Perimeters with Arcs and Straight Lines
For a perimeter, go round the required boundary one piece at a time. A line used to split an area is not automatically part of that boundary.
Separate the required arc from the given arc
Worked example
A semicircle of radius 6 cm has arc AB = 15 cm
AC is its diameter. BX is perpendicular to AC. Find the perimeter of region BXC. (9709/11 May/Jun 2020 Q8)
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The boundary is arc BC, then CX, then XB. Radius OB helps find the lengths but is not on this boundary.
- The given arc AB gives rad. A semicircle has angle , so set .
- The required arc is . In right triangle OXB, sine gives and cosine gives .
- Subtract to get the remaining straight length: .
- Add all three pieces: .
Answer
Trace a different boundary
Your turn
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For OA = 6 cm and OT = 10 cm, find the perimeter of the shaded region ABT. Do not include OA.
Show worked answer
Its boundary is AT + TB + arc BA. Pythagoras gives cm; cm. The arc is cm.
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Answer