Task restatement (in one sentence): We have a circle with centre O. Let A and B be two points on the circle so that triangle OAB is an isosceles right triangle (angle AOB = 90°). Construct the semicircle with diameter AB on the side away from O; the part of that semicircle lying outside the original circle is the lune. Show the area of this lune equals the area of triangle OAB.
- Label and set notation.
Let the radius of the original circle be R, so OA = OB = R. Angle AOB = 90° because triangle OAB is isosceles with right angle at O.
- Compute the area of triangle OAB.
Triangle OAB is a right triangle with legs OA and OB both equal to R. So
Area(triangle OAB) = (1/2) * OA * OB = (1/2) * R * R = R^2 / 2. - Find the semicircle's radius and area.
The length of chord AB equals 2R sin(45°) = 2R * (√2/2) = R√2, so AB = R√2. The semicircle constructed on AB as diameter has radius r = (AB)/2 = (R√2)/2 = R/√2.
Area(semicircle) = (1/2) π r^2 = (1/2) π (R/√2)^2 = (1/2) π (R^2 / 2) = (π R^2) / 4. - Compute the area of the part of the semicircle that lies inside the original circle (a circular segment).
The portion of the original circle cut off by chord AB (the arc AB of the original circle together with chord AB) is a circular sector AOB minus triangle OAB. The sector AOB has central angle 90°, so
Area(sector AOB) = (90/360) * π R^2 = (π R^2) / 4.
So the area of the circular segment (the part of the original circle bounded by chord AB and arc AB) equals
Area(segment) = Area(sector AOB) - Area(triangle OAB) = (π R^2)/4 - (R^2)/2. - Subtract to get the lune area.
The lune is the part of the semicircle outside the original circle, so
Area(lune) = Area(semicircle) - Area(segment inside original circle)
= (π R^2)/4 - [ (π R^2)/4 - (R^2)/2 ]
= R^2 / 2. - Conclusion.
Area(lune) = R^2 / 2 = Area(triangle OAB). Thus the lune has the same area as the isosceles right triangle OAB, as required.
Teacher comments and rubric (firm, high-expectation tone)
Listen carefully. I expect a neat diagram, every step justified, and no skipped algebra. You must label the centre O and points A and B, write the radius R, and show how you found AB and the semicircle radius. If you write an answer without showing how AB = R√2 or without computing the circular segment as sector minus triangle, I will not accept it.
Marking rubric (total 10 marks)
- Diagram and labels: 2 marks — clear centre O, points A and B, chord AB, semicircle drawn on AB.
- Triangle area computed correctly: 2 marks — correct use of legs R and formula 1/2 * R * R.
- Semicircle area computed correctly: 2 marks — show AB = R√2, find r = R/√2, area = (π R^2)/4.
- Circular segment computed correctly: 2 marks — sector area (π R^2)/4 minus triangle area R^2/2.
- Final algebra and conclusion: 2 marks — subtract to get R^2/2 and explicitly conclude areas are equal.
Feedback style (no excuses): If you missed marks, it will be because you omitted an essential justification or the diagram was unclear. Rewrite until every step is present. Show every calculation; be precise with units (R^2), and box your final equality: Area(lune) = Area(triangle OAB) = R^2/2.
ACARA v9 mapping (conceptual, for teacher planning)
This task develops geometric reasoning about areas of circles, sectors and segments and connects them with triangle area — appropriate for middle secondary years (age 13). It supports curriculum aims: investigate properties of circles, use area formulas for circles and sectors, and justify geometric results with clear algebraic reasoning.