Problem 27. To bisect any given triangle ABC by a line drawn parallel to one of its sides AB. E B 1. Bisect one side as AC in D (Pr. 1), produce the bisecting line towards E, and make DE equal to DA or DC. 2. From C, with radius CE, describe an arc meeting AC in F. 3. From F, draw a line FG parallel to AB (Pr. 9), and meeting BC in G. Then the given triangle A BC will be bisected by the line FG. Problem 28. To construct an isosceles triangle on a given base AB, and having a given vertical angle C. F 1. From the angular point C as centre, and with any radius, cut the sides of the angle in D and E, and join DE. 2. At points A and B, make angles equal to the angles at D and E respectively (Pr. 10). 3. Produce the lines forming the angles to meet in F, and FAB will be the required isosceles triangle. Problem 29. To construct an isosceles triangle having its base AB and its altitude CD given. A 1. Bisect the base AB in E (Pr. 1), and from the point of bisection, E, mark off the given altitude CD on the line in the point F. 2. Join FA and FB. Then FAB is the required isosceles triangle. Problem 30. To construct an isosceles triangle on a given base AB, having a vertical angle of 90'. 90 45° 45. 1. At point A, make a right angle (Pr. 2), bisect it (Pr. 6), thereby making BAC equal to 45°. 2. At point B, make an angle ABC equal to angle BAC (Pr. 10). Then ABC will be the isosceles triangle required, and having its vertical angle ACB of 90°. Problem 31. To construct an isosceles triangle on a given base AB, its vertical angle containing a given required number of degrees say in this case 221°). E B 1. Construct an angle CDE containing the required number of degrees, viz. 22} (Pr. 6), and draw a chord to the arc CE. At points A and B in the given line AB, construct angles equal to that at C or E (Pr. 10). 3. Produce the lines completing the angles from A and B until they meet in F. Then FÀB is the required isosceles triangle. Problem 32. To construct an isosceles triangle, one of the equal sides AB, and one of the equal angles C, being given. 1. Draw DE of unlimited length. 2. Make angle EDF equal to angle C (Pr. 10), and make DF equal to AB. LA E 3. With centre F, and radius FD, describe arc DG. its two sides FD, FG equal to the given line AB, and its Problem 33. To construct an isosceles triangle, having its base AB and its perimeter CD given. i. Bisect AB and CD in the points E and F (Pr. 1), and produce the bisecting line through F indefinitely towards G. |