Show how to construct a triangle given the length of one side, the size of an adjacent angle and the length of the median from that angle
1. Draw the base of the triangle with the given length, call it AB.
2. Draw a line at the prescribed angle to the base, intersection the base at point A. Produce the line above and below the base. The third vertex above the base will be called C, and that below the base will be called C'.
3. Draw a circle, centred at B, radius equal to length of median.
4. From point B, draw a perpendicular to line C'C, cutting it at D.
5. Bisect segment BD and call the mid-point E.
6. Draw a line parallel to C'C through E intersecting the circle at two points M (above AB) and M' (below AB). M and M' are the mid-points of side BC and BC' respectively of the triangles to be constructed.
7. Join BM and produce to intersect C'C at C. Similarly, join BM' and produce to intersect C'C at C'.
8. The required triangles are ABC and ABC'.
Note: The above instructions assume ∠ABC to be acute. If ∠ABC is obtuse, the instructions may vary slightly, but the principle is the same.
To construct a triangle given the length of one side, the size of an adjacent angle, and the length of the median from that angle, follow these steps:
1. Draw a base: Start by drawing a line segment to represent the known side of the triangle. This will be the base of the triangle.
2. Measure the angle: Use a protractor to measure the given adjacent angle at one endpoint of the base. Mark the angle with the appropriate measurement.
3. Construct the median: From the endpoint of the base where the angle is located, draw a line segment that is perpendicular to the base. The length of this line segment should match the given length of the median.
4. Bisect the median: Using a compass, find the midpoint of the median line segment. Draw a small arc above and below the midpoint on the median line.
5. Connect the arcs: With the compass set to any convenient radius, draw an arc from the endpoint of the base where the angle is located. Ensure that the arc intersects both arcs on the median line.
6. Connect the points: Using a ruler, draw a straight line from the endpoint of the base to the intersection point of the two arcs. This line will be the final side of the triangle.
7. Complete the triangle: Connect the other endpoint of the base to the endpoint of the final side using a ruler. This will close the triangle.
You have now constructed a triangle given the length of one side, the size of an adjacent angle, and the length of the median from that angle.