Ex. 5. Which passes through two given points and has its center on a given line. Ex. 6. Which touches three given lines, two of which are parallel. Ex. 7. Which passes through a given point A and touches a given line BC at a given point B. [SUG. Draw AB and at B construct a L to BC.] Ex. 8. Which touches a given line and also touches a given circle at a given point 4. -B Ex. 9. Which touches a given line AB at a given point A and touches a given circle. EXERCISES. GROUP 28 PROBLEMS SOLVED BY VARIOUS METHODS Ex. 1. Through a given point draw a line which shall cut two given intersecting lines so as to form an isosceles triangle. Ex. 2. Construct an isosceles triangle, given the altitude and one leg. Ex. 3. In a given circumference find a point equidistant from two given intersecting lines. Ex. 4. Draw a circle which shall touch two given intersecting lines, one of them at a given point. Ex. 5. Draw a line which shall be terminated by the sides of a given angle, shall equal a given line, and be parallel to another given line. Ex. 6. Construct a triangle, given one side, an adjacent angle, and the difference of the other two sides. Ex. 7. Find a point in a given circumference at a given distance from a given point. Ex. 8. Construct a parallelogram, given a side, an angle, and a diagonal. Ex. 9. Through a given point within an angle, draw a straight line terminated by the sides of the angle and bisected by the given point. [SUG. Draw a line from the vertex of the angle to the given point and produce it its own length through the point.] Ex. 10. Construct a triangle, given the vertex angle and the segments of the base made by the altitude. [SUG. Use Art. 291.] Ex. 11. Construct an isosceles triangle, given the angle at the vertex and the base. Ex. 12. Draw a circle with given radius which shall touch a given circle at a given point. Ex. 13. Construct a right triangle, given the hypotenuse and the altitude upon the hypotenuse. Ex. 14. Construct a triangle, given the base and the altitudes upon the other two sides. [SUG. Construct a semicircle on the given base as a diameter.] Ex. 15. Find a point in one side of a triangle equidistant from the other two sides. Ex. 16. Construct a triangle, given the altitude and the angles at the extremities of the base. Ex. 17. Construct a rhombus, given an angle and a diagonal. Ex. 18. Draw a circle which shall pass through two given points and have its center equidistant from two given parallel lines. Ex. 19. Construct a triangle, given one side, an adjacent angle and the radius of the circumscribed circle. Ex. 20. In a given circle draw a chord equal to a given line and parallel to another given line. [SUG. Find the distance of the given chord from the center, by constructing a right triangle of which the hypotenuse and one leg are given.] Ex. 21. Construct a triangle, given an angle, the bisector of that angle, and the altitude from another vertex. Ex. 22. Find the locus of the points of contact of tangents drawn from a given point to a series of circles having a given center. Ex. 24. Given a line AB and two points C and D on the same side of AB; find a point P in AB such that CP+PD shall be a minimum. Ex. 25. Draw a common external tangent to two given circles. Ex. 26. Draw a common internal tangent to two given circles. BOOK III PROPORTION. SIMILAR POLYGONS THEORY OF PROPORTION 298. Ratio has been defined, and its use briefly indicated in Arts. 245, 246. 299. A proportion is an expression of the equality of two or more equal ratios. α b d' As, or a b This reads, "the ratio of a to b equals the ratio of c to d," or, "a is to b as c is to d." 300. The terms of a proportion are the four quantities used in the proportion. In a proportion the antecedents are the first and third terms; the consequents are the second and fourth terms; the extremes are the first and last terms; the means are the second and third terms. A fourth proportional is the last term of a proportion (provided the means are not equal). Thus, in a b=c : d, d is a fourth proportional. 301. A continued proportion is a proportion in which each consequent and the next antecedent are the same. Thus, a b b:cc: d=d: e is a continued proportion. A mean proportional is the middle term in a continued proportion containing but two ratios. L : (177) A third proportional is the last term in a continued proportion containing but two ratios. Thus, in a b=b: c, b is a mean proportional, and c is a third proportional. PROPOSITION I. THEOREM 302. In any proportion, the product of the extremes is equal to the product of the means. Given the proportion a: b=c d. 303. The mean proportional between two quantities is equal to the square root of their product. Given the proportion a: bbc. (in any proportion, the product of the extremes equals the product of the means). Q. E. D. Ex. 1. Find the fourth proportional to 2, 3 and 6; also to 3, 1, . Ex. 2. Find the mean proportional between 3 and 6. Ex. 3. Find the third proportional to 3 and 5. |