Let's break it down step by step:
(a) Drawing triangle ABC:
1. Use a ruler to draw line segment AB measuring 11 cm.
2. With point A as the center, use a compass to draw an arc with radius 8 cm.
3. With point B as the center, use a compass to draw another arc with radius 5.6 cm, ensuring it intersects the first arc.
4. Label the intersection point of the arcs as point C.
5. Connect points C to A and C to B to complete triangle ABC.
(b) Constructing the bisectors of two angles:
6. To construct the angle bisector of ∠A, place the compass on point A and draw an arc that intersects AB and AC. Label these intersections as D and E respectively.
7. Without altering the compass width, draw arcs from points D and E that intersect at a point inside the triangle. Label this intersection point as P.
8. Draw a line from point A through point P. This is the angle bisector of ∠A.
9. Repeat the same process to bisect either ∠B or ∠C.
10. The intersection of these two bisectors is point R.
(c) Drawing the perpendicular from R to AB:
11. Place the compass on point R and draw an arc that crosses AB at two points. Label these points as F and G.
12. Without changing the width of the compass, draw arcs from F and G that intersect each other. Label this intersection point as H.
13. Draw a line from point R through point H. Let the intersection of this line with AB be M. This is the perpendicular from R to AB.
(d) Drawing a circle with center R and radius RM:
14. Place the compass on point R and adjust it to the length of RM.
15. Draw a circle with center R and radius RM.
(e) Calculating the area of the circle:
16. Given that RM is the radius, we can use the formula for the area of a circle: [tex]\( \text{Area} = \pi \times \text{radius}^2 \)[/tex]
17. From the already known solution, the radius (RM) is calculated to be 8 units.
Thus, the area of the circle is:
[tex]\[
\text{Area} = \pi \times 8^2 = \pi \times 64 \approx 201.06 \text{ square units}
\][/tex]
So, the detailed solution follows the aforementioned steps to draw, bisect angles, construct perpendiculars, and finally calculate the area of the circle.
