Ex. 692. Touching a given circle at a given point and passing through another given point without. Ex. 693. Touching a given line and a given circle at a given point. (HINT. Draw a tangent to the circle at the given point.) * Ex. 694. Construct a triangle having given a, ha, A. Ex. 695. Construct a triangle having given a, mɑ, A. 268. No general method can be given for the solution of exercises; a great many, however, can be solved: (1) By a gradual putting together of the given parts. (2) By means of an analysis. (3) By means of loci. (247) MISCELLANEOUS EXERCISES Construct an isosceles triangle, having given : Ex. 696. The base and the altitude upon an arm. Ex. 697. The altitude upon the base and the vertex angle. Ex. 698. The vertex angle and the sum of one arm and the base. Ex. 699. The perimeter and the base angles. Construct a right triangle, having given: Ex. 700. One acute angle and the altitude upon the hypotenuse. Ex. 701. The altitude upon the hypotenuse and one of the segments of the hypotenuse. Ex. 702. The sum of the arms and one acute angle. Ex. 703. To find a point in one side of a triangle which is equidistant from the other two sides. Ex. 704. Find the locus of the vertex of a right triangle, having a given hypotenuse. Ex. 705. In one side of a quadrilateral to find a point equidistant from the ends of the opposite side. Ex. 706. From a point P, in the circumference of a circle, to draw a chord, having a given distance from the center. Ex. 707. In a given circle, to draw a diameter having a given distance from a given point. Ex. 708. Through a given point to draw a line, having a given distance from another point. Ex. 709. Through two given points in a circumference, to draw two equal parallel chords. Ex. 710. Trisect a given straight angle. Ex. 713. Through a given point, to draw a line cutting off equal lengths on the sides of a given angle. Ex. 714. Through a given point, to draw a line making a given angle with a given line. Ex. 715. Through a given point, to draw a line of given length termiating in two given parallel lines. Ex. 716. Through a given point, to draw a line making equal angles with two given lines. * Ex. 717. To bisect an angle formed by two lines, without producing them to their intersection. To construct a triangle, having given : (Note 244) Ex. 732. The difference between the diagonal and the side. Ex. 733. The sum of the diagonal and the side. To construct a rectangle, having given : Ex. 735. One side and the angle formed by the diagonals. Ex. 738. The perimeter and one diagonal. Ex. 739. One angle and a diagonal. Ex. 740. The altitude and the base. Ex. 741. The altitude and one angle. Ex. 742. Two adjacent sides and one altitude. Ex. 743. Ex. 744. One side and two diagonals. Ex. 745. One side, one angle, and one diagonal. Ex. 746. The diagonals and the angle formed by the diagonals. 269. In the analysis of a problem relating to a trapezoid, draw a line through one vertex, 4, either parallel to the opposite arm, DC, or parallel to a diagonal, DB. To construct a trapezoid, having given: Ex. 747. The four sides. Ex. 748. The bases and the base angles. Ex. 749. The bases, another side, and one base angle. Ex. 750. The bases and the diagonals. I Ex. 751. One base, the diagonals, and the angle formed by the diagonals. Ex. 752. To draw a common external tangent to two given circles. Ex. 753. To draw a common internal tangent to two given circles. Ex. 754. About a given circle, to circumscribe a triangle, having given the angles. Ex. 755. Find the locus of the mid-points of the secants that pass through a given point without a circle. Ex. 756. In a given circle, to inscribe a triangle, having given the angles. * Ex. 757. From a given point in a circumference, to draw a chord that is bisected by a given chord. Ex. 759. Given two points, A and B, on the same side of a line, CD. To find a point, X, in CD, such that ▲ AXC = 2 BXD. [See practical problems 44-53, pp. 293 and 294.] *A BOOK III PROPORTION. SIMILAR POLYGONS 270. DEF. A proportion is a statement of the equality of NOTE. The statements α с = = c: d. and abc:d, are absolutely identical. b d Hence, if a hypothesis states a this statement in the form : α C b d Similarly, if we have to prove a : b b c : d, we may in the proof employ without assigning a special reason. 271. DEF. The first and the fourth terms of a proportion are called the extremes, the second and the third, the means. 272. DEF. The first and the third terms are called the antecedents, the second and the fourth the consequents. Thus, in the proportion, a : b = c :d, a and d are the extremes, b and c the means, a and c the antecedents, and b and d the consequents. 273. DEF. When the means of a proportion are equal, either mean is said to be the mean proportional between the first and the last terms, and the last term is said to be the third proportional to the first and the second terms. Thus, in the proportion, a:bb: c, b is the mean proportional between a and c, and c is the third proportional to a and b. 274. DEF. The last term is the fourth proportional to the first three. Thus, in the proportion, a : b = c : d, d is the fourth proportional to a, b, and c. |