Question

the following theorems may be used to answer the question below: power if the point, law if cosines, extended law of sines, incenter, excenter, orthocenter, and circumcenter. Note that the distance between O to A is 5.5cm, G to O is 1.8 cm and G to A is 6.1cm

6, Below you are given the vertex A, centroid G, and the circumcenter O of △ABC. Construct the other two vertices B and C ●A

Answer 1

1. Centroid is the point of intersection of medians. Also, Centroid divides the median, say median AD, in the ration 2:1 i.e. AG: GD = 2:1.

2. Draw a line l passing through A and G.

3. Find the mid-point of the segment AG: Open the compass more than half the length of segment AG. With A as centre, draw two arcs, one on either side of the segment AG. With the same compass length, draw two more arcs with G as center, such that these arcs intersect the old arcs, say at points P and Q. Draw a line passing through points P and Q. This line intersects the segment AG at point M, say. Then point M is the mid point of segment AG.

4. Open the compass equal to the half the length of line segment AG i.e. equal to AM = MG. With G as center draw an arc intersecting the line l at point D ( the mid-point of side BC of the triangle)

5. Now, circumcenter is the point of intersection of perpendicular bisectors of the sides. Therefore, to find side BC, draw a line m passing through O and D.

6. Draw another line n perpendicular to line m at D: With D as center, draw 180 degree angle or semi-circle on line m such that it intersects the line m at points R and S. With R as center, draw an arc. With the same radius and S as centre, draw another arc intersecting the previous arc at point T, say. Draw line n passing through T and D.

7. With O as center and radius equal to distance OA, draw two arcs intersecting the line n at points B and C, the other vertices of the triangle ABC.

8. Join A to B, and A to C. This completes the triangle ABC.

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