For each of the following constructions:
6. Construct a triangle, given the measure of one angle, the length of an adjacent side, and the length of the altitude to that side.
Solution:
b) Let angle A, AB and CD be the given elements. Vertex C is on the side of A other than AB and also on the line parallel to AB at a distance equal to CD.
c) The solution is unique.
8. Construct a triangle, given the length of one side and the lengths of the medians to the other two sides.
Solution:
b) Let BC, CC' and BB' be the given elements. The centroid D can be determined as being 2/3 of CC' from C and 2/3 of BB' from B. C' and B' can then be found on CD and BD respectively. A is then determined as the intersection of CB' and BC'.
c) If the centroid can not be determined there will be no solution. If CB' and BC' are parallel there will be no solution. Otherwise the solution is unique.
10. Construct a triangle, given the length of one side, the length of the median to that side, and the circumradius.
Solution:
b) Let AB and CC' be the known lengths. The center of the circumcircle can be found using the circumradius from points A and B. Thus, the circumcircle can be constructed, and vertex C must be on it. From the midpoint of AB a circle with radius CC' can be drawn. C will be a point of intersection of this circle with the circumcircle.
c) If the circumradius is less than half of AB, no circumcircle can be drawn and there will be no solutions. If the two circles do not intersect, C can not be determined and again no solution will exist. If the two circles intersect in one or two points, there will be one or two solutions. If the circles intersect in three points, they are the same circle and C could be any point other than A or B on this circle, giving an infinite number of solutions. (Note: the answer in the back of the text is incorrect, it leaves out this possibility.)
12. Construct a circle with a given radius tangent to a given line and tangent to a given circle.
Solution:
b) Let the circle with center O, and the line l be given. The circle with center O' is to be constructed. Since the radius of this circle is known, O' must be on a line parallel to l, at a distance equal to this radius. Since the circle is to be tangent to the given circle, O' is also on a circle concentric to the given circle with radius equal to the radius of the given circle plus the radius of the O' circle.
c) Because there are two lines parallel to l at the given distance and since O' is determined as the intersection of one of these lines with a circle, there can be 0, 1, 2, 3 or 4 solutions.
17.Construct a triangle, given the length of one side, the length of the median to that side, and the length of one other median.
Solution:
b) Let BC be the given side, AA' the median to that side and CC' the other median. The centroid D can be located at 1/3 of AA' from the midpoint of BC and 2/3 of CC' from C. A can then be located on DA'.
c) If the centroid D can not be determined, there will be no solution. Otherwise the solution is unique.