Miscellaneous Exercises Ex. 152. The vertices of quadrilateral ABCD are joined to a point O lying outside the quadrilateral. Points A', B', C', and D' are taken on OA, OB, OC, and OD, respectively, so that A'B' || AB, B'C' || BC, and C'D' CD. Prove A'D' || AD. Ex. 153. Two circles are tangent externally at point C. Through C, a straight line is drawn, meeting the first circle at A and the second at D; another straight line through C meets the first circle at B and the second at E. Prove AC: CD BC: CE. = Suggestion. - Draw the common tangent at C, and also chords AB and ED. Ex. 154. If P and S are two points on the same side of line OX such that the perpendiculars PR and ST drawn to OX have the same ratio as OR and OT, then points O, P, and S lie in a straight line. Suggestion.-Prove 4 ROP = LTOS by proving ▲ OPR~ A OST. Ex. 155. If two parallels are cut by three or more straight lines passing through a common point, the corresponding segments are proportional. A' ი Ex. 156. If three transversals intercept proportional lengths on two parallels, the transversals meet at a point. Suggestion. Let A'A and B'B meet at O and draw OC and OC'; then prove A OBC and OB'C' similar. (Fig. Ex. 155.) Ex. 157. Derive a formula for the altitude to the base of an isosceles triangle if the base is b and the equal sides are each a. By means of the formula determine the altitude when: (a) a = 12 and b = 6; (b) a = 15 and b = 7. Ex. 158. Find the length of the common external tangent to two circles whose radii are 11 and 18, if the distance between their centers is 25. Suggestion.-See the figure for Problem 2, § 236. Ex. 159. If BE and CF are the medians drawn from the extremities of the hypotenuse of right triangle ABC, prove 4 BE2 + 4 CF2 = 5 BC2. Ex. 160. Prove that the projections of two parallel sides of a parallelogram upon either of the other sides are equal. Ex. 161. BC is the base of an isosceles triangle ABC inscribed in a circle. If a chord AD is drawn, cutting BC at E, prove AB2 = AE2 + BE × CE. Suggestions.-1. The proof is like that for § 318. 2. Prove A ABD ~ ▲ ABE. Ex. 162. Prove that the non-parallel sides of a trapezoid and the line joining the middle points of the parallel sides, if extended, meet in a coinmon point. Review Questions 1. What is meant by taking a proportion by (a) inversion? (b) composition? (c) alternation? 2. Complete the following theorem: "If the product of two numbers equals the product of two other numbers, one pair, 3. Define: (a) mean proportional; (b) fourth proportional; (c) similar polygons; (d) ratio of similitude. 4. Are mutually equiangular triangles similar? Are mutually equiangular polygon's similar? 5. State all of the theorems by which two triangles can be proved similar. 6. How do you select the homologous sides of similar triangles? What do you know about them? 7. What do you know about the ratio of homologous altitudes of similar triangles ? What about the ratio of the perimeters of similar triangles ? 8. What is the Pythagorean theorem ? 9. Find the mean proportional between 5 and 15. Construct the mean proportional between segments r and s. 10. What are the two conclusions which follow from the hypothesis that the altitude is drawn to the hypotenuse of a right triangle ? BOOK IV AREAS OF POLYGONS 319. A polygon, being a closed line (§ 7), incloses a limited portion of the plane. In measurement theorems, the words "rectangle," "parallelogram,” "polygon," etc., mean the surface within the figure mentioned. 320. The Area of the surface within a closed line is the ratio of the surface to the unit of surface measure. Thus, in the adjoining figure, if the unit of surface is one small square, the area of the rectangle is 30. It has become customary, when speaking of the area of a figure, to mention at once the unit of surface; thus, in the foregoing example, it is customary to say that the area is 30 small squares. Remember, however, that the area is 30. A 321. The usual Unit of Surface is a square whose side is some linear unit: as, a square inch or a square centimeter. In this text, it will be assumed that the unit always is such a square unit. Ex. 1. In the following figures, assume that the unit of surface is a small square. (a) What is the exact area of Figs. 1 and 2? (b) What is the approximate area of Figs. 3, 4, and 5? (Include the square in the area if half or more than half of it lies within the figure; do not include it otherwise.) 322. Two limited portions of a plane are Equal (=) if their areas are equal when they are measured by the same unit. Since the test of the equality of two figures is the equality of two numbers, the usual axioms apply when equal figures are added or sub-' tracted, or when they are multiplied or divided by the same number. Thus, if equal figures are added to equal figures, the sums are equal; also halves of equal figures are equal. Ex. 2. Of the figures in Ex. 1, are any two or more equal ? 323. Two congruent figures are necessarily equal, but two equal figures are not necessarily congruent. Also, two figures which consist of parts which are respectively congruent are equal. Thus, the parallelogram and the kiteshaped figure made from it by placing the two triangles together as in the figure adjoining are equal. B Ex. 3. If E is the mid-point of one of the nonparallel sides of trapezoid ABCD, and a parallel to AB drawn through E meets BC extended at F and AD at G, prove that parallelogram ABFG is equal to trapezoid ABCD. A Suggestion.-Prove ▲ CEF ≈▲ GED, and apply § 323. Ex. 4. In the adjoining figure, D and E are the mid-points of AB and AC; AJ1 BC; BF 1 DE extended at F; CG 1 DE extended at G. Prove that ▲ ABC equals □ BFGC. Suggestion. Prove ▲ BDF ≈ ▲ DAH, and A CEGA AEH. 11 = B Ex. 5. In the adjoining figure, E and F are the mid-points of sides AB and CD of trapezoid ABCD; XY and Z W are drawn through E and F respectively 1 AD, meeting BC extended at X and W respectively. Prove that XYWZ = ABCD. Ex. 6. Let K be the mid-point of side BC and H the mid-point of side AD of ABCD; let FE, drawn through the mid-point G of KH, intersect BC and AD at F and E respectively. Prove that FE divides ABCD into two equal quadrilaterals. Note. Supplementary Exercises 1-3, p. 294, can be studied now. MEASUREMENT OF RECTANGLES 324. The Dimensions of a rectangle are the Base and Altitude. 325. Area of a Rectangle. (Informal treatment.) If the base of a rectangle measures 6 and its altitude 5 linear units, the area is evidently 6 × 5 or 30 surface units. If the base measures 6 units and the altitude measures 3 units, the area is evidently 6 x 3.5 or 21 surface units. These two examples suggest the theorem: The number of surface units in the area of a rectangle is the product of the number of linear units in its base and the number in its altitude. More briefly, this theorem is expressed: the area of a rectangle is the product of its base and its altitude. The theorem is proved in the following three propositions. 326. Comparison of Rectangles. Rectangles may be compared without computing their areas. Two rectangles having equal bases and altitudes are equal, for it is evident that they can be made to coincide by superposition. |