Using a pair of compasses and a ruler only, construct i.)∆ABC with|AB|=8cm, angle ABC=30°and angle CAB=45°.ii.)a perpendicular from the point C to meet the line AB at P. b.)i.) Extend line CP to the point D such that|BC|=|BD|.ii.) join A to D and B to D iii.) measure|AC|and|AD|
To construct triangle ABC:
1. Using a ruler, draw a line segment AB of length 8 cm.
2. With A as the center, draw an arc on the same side as B, intersecting AB at point C.
3. With C as the center, draw an arc to create an angle of 30 degrees. The arc should intersect the previous arc drawn at point D.
4. Connect points A and D with a straight line segment.
5. With B as the center, draw an arc on the opposite side, intersecting AD at point E.
6. Use a ruler to connect points B and E, creating the line segment BE.
7. Finally, label the points as A, B, and C.
To construct a perpendicular line from point C to line AB:
8. With C as the center and any radius greater than half of AB, draw two arcs intersecting line AB at points F and G.
9. With the same radius, draw arcs intersecting the arcs drawn in step 8 at points H and I.
10. Connect points G and I with a straight line.
11. The line CI is the perpendicular from point C to line AB, and it meets line AB at point P.
To extend line CP to create a point D such that BC = BD:
12. Use a compass to measure the length of BC from point C.
13. Without changing the compass width, place the compass point on point P and draw an arc away from line CP.
14. Keeping the same compass width, place the compass point on point C and draw another arc intersecting the first arc. Label this intersection point as D.
15. Connect points C and D with a straight line.
To measure |AC| and |AD|:
16. Use a ruler to measure the length of line segment AC, which is the distance between points A and C.
17. Use a ruler to measure the length of line segment AD, which is the distance between points A and D.
To construct the given figure using a pair of compasses and a ruler, follow these steps:
i.) Constructing ∆ABC:
1. Draw a line segment AB of length 8 cm using the ruler.
2. Place the point of the compass on point A and draw an arc cutting the line segment AB.
3. With the same compass width, place the point on point B and draw another arc cutting the previous arc.
4. Label the points where the intersection of the arcs occur as C.
ii.) Constructing a perpendicular from C to meet AB at P:
5. Open your compass to a distance greater than half of AB.
6. With the point of the compass on point C, draw an arc that intersects AB at two different points.
7. Without changing the compass width, place the point of the compass on one of the intersection points on AB and draw another arc that intersects the previous arc.
8. Draw a line segment between point C and the intersection point of the two arcs. Label this point as P.
b.) Constructing ∆BCD:
9. Draw a line segment between points C and P using the ruler.
10. Open your compass to a distance equal to the length of BC.
11. With the point of the compass on point B, draw an arc that intersects the line CP.
12. Without changing the compass width, place the point on point C and draw another arc that intersects the previous arc.
13. Label the points where the intersection of the arcs occur as D.
ii.) Joining AD and BD:
14. Draw a line segment between points A and D.
15. Draw a line segment between points B and D.
iii.) Measuring |AC| and |AD|:
16. With a ruler, measure the length of line segment AC to determine |AC|.
17. With a ruler, measure the length of line segment AD to determine |AD|.