1a) Consider the triangle OAB
i) Find the area of the triangle
ii) Find the angle x
b) Find the area of the sector OAPB
c) Find the area of the shaded region.
d) Find the perimeter of the shaded region.
The diagram shows the construction of the Druid Temple stone circle in Scotland which is an example of an Egg-shaped circle of type 1.
Two identical right-angled triangles, ABC and ABD, are placed base to base. A semi-circle PXQ of radius r and centre A is drawn.
An arc with centre D is drawn from P until it meets the line DB at the point U.
An arc with centre C is drawn from Q until it meets the line CB at the point V.
An arc with centre B is drawn from U to V.
For the circle at Druid Temple:
AB= 3 megalithic yards; AC= 4 megalithic yards; r=7 megalithic yards.
Calculate the length of the perimeter of the stone circle.
Use Heron's formula to find the area of the triangle. Then use the Law of Cosines to find angle x. Then use that angle x is a fraction of 360 degrees and hence your sector is the same proportion to the area of a circle.
c) and d) make no sense as there is no shaded region.
-Dan
The area of the triangular portion is easy, just use the Pythagorean theorem to find the second leg of the triangle. Then if we calculate angle BOA (from some simple trig), which is supplementary to angle COB, that tells you what fraction of the circle the sector is.
-Dan
I've modified your sketch a little bit:
1. Split the triangle OAB into 2 right triangles.
2. Calculate the length of the red line ($\displaystyle \sqrt{28}$)
3. Both right triangles form a rectangle whose area is:
$\displaystyle A_{OAB} = 6 \cdot \sqrt{28}=12 \cdot \sqrt{7}\approx31.75\ cm^2$
4. Use the Sine function to calculate the value of $\displaystyle \frac12 x$
5. The area of the sector is calculated by:
$\displaystyle \frac{A_{sector}}{\pi \cdot 8^2} = \frac{x}{360^\circ} $
I can't help you with the last two questions because there isn't any shaded region