In the diagram, $$\(A B C\)$$ is an isosceles triangle with $$\(A B=B C=r \mathrm{~cm}\)$$ and angle $$\(B A C=\theta\)$$ radians. The point $$\(D\)$$ lies on $$\(A C\)$$ and $$\(A B D\)$$ is a sector of a circle with centre $$\(A\)$$. Express the area of the shaded region in terms of $$\(r\)$$ and $$\(\theta\)$$. .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... .................................................................................................................................................... ....................................................................................................................................................
Exam No：9709_w20_qp_12 Year：2020 Question No：8(a)
Answer:
Use of correct formula for the area of triangle \(A B C\)
\(\frac{1}{2} r^{2} \sin (\pi-2 \theta)\) or \(\frac{1}{2} r^{2} \sin 2 \theta\) or \(2 \times \frac{1}{2} r \times r \cos \theta \times \sin \theta\) or
\(
2 \times \frac{1}{2} r \cos \theta \times r \sin \theta
\)
\([\) Shaded area \(=\) triangle \(-\) sector \(]=\) their triangle area \(-\frac{1}{2} r^{2} \theta\)
\(\frac{1}{2} r^{2} \sin (\pi-2 \theta)\) or \(\frac{1}{2} r^{2} \sin 2 \theta\) or \(2 \times \frac{1}{2} r \times r \cos \theta \times \sin \theta\) or
\(
2 \times \frac{1}{2} r \cos \theta \times r \sin \theta
\)
\([\) Shaded area \(=\) triangle \(-\) sector \(]=\) their triangle area \(-\frac{1}{2} r^{2} \theta\)
Knowledge points:
1.4.2 use the formulae in solving problems concerning the arc length and sector area of a circle (Including calculation of lengths and angles in triangles and areas of triangles.)
Solution:
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