414. Problem. To construct the fourth proportional to three given lines. Construction. From any point P draw two lines, PE and PF, making any convenient angle. Lay off PG=a, GH =b, and PI=c. Draw GI, and draw HJ || GI, meeting PE at J. Then d is the required fourth proportional. Proof. In APHJ, GI || HJ, 1. Find the fourth proportional to three lines that are 1 in., 12 in., and 14 in. long, respectively. Solve the exercise both by geometry and by algebra and compare the results. 2. Find the value of x in 15x=25:20. Use both algebra and geometry and compare results. 3. Give algebraic and geometric solution for x, if 36 50 50 x 4. A given rectangle has a base b and an altitude a. base of an equivalent rectangle that has a given altitude c. Construct the 5. Construct the altitude of a rectangle of given base b, that is equivalent to a square of given side a. 415. Problem. To divide a given straight line into parts that are in a given ratio. Construction. Draw AC making a convenient angle with AB. On AC lay off AD=a and DE=b. Draw EB and through D draw DF || EB. Then AB is divided in the required ratio by F. Proof. In AABE, DF || EB. 416. Problem. To divide a given straight line into parts proportional to any number of given lines. Given the lines AB, a, b, c, and d. Required to divide AB into parts proportional to a, b, c, and d. Construction. Draw AC making any convenient angle with AB. On AC lay off successively a, b, c, and d as shown. Draw GB and draw lines parallel to GB through F, E, and D. Then AB is divided as required. Why? EXERCISES 1. Divide a line 4 inches long into parts that are in the ratio of 7:9. Solve both by algebra and by geometry and compare results. 2. In drawing to scale, lines are divided proportionally. Find the length of each side of these figures on a scale of 80 miles to the inch. 3. Construct a scale showing 60 miles to the inch. 4. Construct a scale showing 50 miles to the centimeter. 5. Divide a line 28 ft. in length into parts that are in the ratio of 1:2:34. Solve both by algebra and by geometry and compare results. 6. By a line through a vertex, divide a triangle into parts that are in the ratio of two given straight lines. Into parts that are in the ratio of any number of given lines. § 368 7. Construct a triangle having its sides in the ratio of 23:4, and its perimeter equal to a given straight line. B 8. Through a point P without an angle ABC, draw a line intersecting AB in D and CB in E, so that PD=DE. So that PD: DE as any given ratio. A 9. Through a point P within an angle ABC, draw a line intersecting AB in D and CB in E, so that DP=PE. So that DP PE as any given ratio. SUGGESTION. Draw PN || AB and lay off NE=BN. Could PN be drawn parallel to BC as well? B N E A 10. ABC is a triangle, and D and E are points in AB and AC, respectively, such that DE is parallel to BC. BE and CD meet at F. Prove that ▲ADF is equiva- 4/ lent to AAEF. D *11. Divide a line 12 in. long into segments that are in the ratio of √2:√3. Construct and compute. Which is the better method? Upon what does the inaccuracy in each depend? 417. Theorem. The bisector of an angle of a triangle divides the opposite side into two parts which are proportional to the adjacent sides. Given AABC, with BD bisecting B Proof. Through C draw CE || DB to meet AB produced at E. Lx=Ly, Lx = Zz, and Zy=Zw. Why? Why? Why? AD AB § 411 Why? EXERCISES 1. In the figure of § 417: (1) Given AB=20, AD=14, DC=12; find BC. 2. The three sides of a triangle are respectively 10, 14, and 22. Find the segments of the sides made by the bisectors of each of the angles. 3. If a straight line bisects both the vertex angle and the base of a triangle, the triangle is isosceles. 4. State and prove the converse of the theorem of § 417. 5. In AABC, BD bisects the exterior angle at B. Show that AD: CD=AB : BC. E C SUGGESTION. Draw CE || DB and prove that CB=EB. SIMILARITY 418. Geometric figures may be equivalent, similar, or equal. Equivalent figures have the same size, similar figures have the same shape, and equal figures have the same size and shape.* Similarity is of great practical importance. By similarity, many computations are shortened to a marked degree. Drawings, in general, represent objects of a size different from that of the drawing, but are made so that the reduction or the enlargement is in the same ratio throughout. An architect in making a plan of a building, constructs the drawing to some scale. A map of a portion of the earth is always drawn to some scale. For instance, if the scale is 1:1,000,000, each line on the surface of the earth is 1,000,000 times as long as it is on the map. 419. Definitions. Two polygons are said to be mutually equiangular if the angles of one polygon, taken in order, are equal respectively to the angles of the other polygon, taken in the same order. The pairs of equal angles are called corresponding angles of the two polygons. Two lines, one in each polygon, having the same relative positions with respect to corresponding angles are called corresponding lines. EXERCISES 1. Make a drawing of your school room on a scale of 1:96, that is, in. to 1 ft. 2. If a polygon of cardboard is held parallel to a wall and between a light and the wall, what will be the shape of the shadow on the wall? 3. Are two circles of the same shape? Two spheres? Two rectangles? Two triangles? Two equilateral triangles? Two isosceles triangles? Two squares? *These statements are not to be considered as definitions. Equal is here used in the sense of congruent. |