sides about the equal angles proportional, that is, AB to BC as ab to bc, and BC to CD as bc to cd, and so on.

72. Thus to ensure the similarity of rectilineal figures we must have two properties, namely, equality of angles and proportionality of sides. Theory demonstrates that if two triangles have one of these properties they will necessarily have the other; and it is easy to test this practically. For example, let two triangles be drawn on paper, such that the sides of one are twice or three times as long as the sides of the other; cut the triangles out, and apply one triangle on the other; it will be found that the corresponding angles are equal. But in the case of rectilineal figures having more than three sides, either of the properties may exist singly without the other. For example, take a square and any rectangle which is not a square; here the angles of one figure are respectively equal to the angles of the other, but the sides are not proportional. Again, take a square and a rhombus; here the sides of one figure are all in the same proportion to the sides of the other figure, but the angles of one figure are not equal to the angles of the other figure.

73. Similar rectilineal figures can always be divided into the same number of similar triangles. Thus, for example, by drawing the straight lines CA, DA, ca, da,

we can divide the five-sided figures of Art. 71 into three pairs of similar triangles.

T. M.

3

74. The statement made in Art. 66 with respect to similar triangles holds for any two similar rectilineal figures; that is, if two straight lines situated in one figure be given, and a straight line corresponding to one of them in a similar figure, the straight line corresponding to the other can be found by Proportion.

75. Similar figures may occur which are bounded by curved lines as well as those which are bounded by straight lines. Thus, two maps of different sizes may represent the same country; the two maps will then be similar. For example, one map may be on the scale of an inch to a mile, and the other on the scale of half an inch to a mile then any line drawn on the first map will be twice as long as the corresponding line drawn on the second map.

76. A good notion of similar figures may be conveyed, by saying that they are exactly alike in form although they may differ in size.

All circles are similar figures.

77. We will now solve some exercises which depend on the similarity of figures.

(1) In the diagram of Art. 73 suppose AE-2 inches, AC 4 inches, ae1 inches: find ac.

(2) In the preceding exercise find the proportion of AD to ad.

Since AE-2 and ae=14, any straight line as AD bears to the corresponding straight line ad the proportion of 2 to 14, that is of 2 to, that is of to, that is of 8 to 5.

(3) In the diagram of Art. 37 if BC=15, and BA = 12, find BD.

The triangles ABC and DBA are similar: thus

(4) ABCD is a trapezoid. The distance of the parallel sides AB and CD is 3 feet; AB=10 feet; DC-6 feet. Let AD and BC produced meet at O. It is required to find the perpendicular distance of O from DC.

Draw DG perpendicular to AB, and DH parallel to BC. Then BH=DC: thus AH-10-6=4. Also DG=3. Now the triangles ADH and DOC are similar. Therefore by Art. 35,

AH: GD: DC: required distance.

EXAMPLES. VI.

1. In the diagram of Art. 36 if AD=5 inches, DE=4, and AB=7: find BC.

2. The side of an equilateral triangle is 2 feet 6 inches: find the height.

3. The shadow of a man 6 feet high standing upright was measured and found to be 8 feet 6 inches: the shadow of a flag-staff, measured at the same time, was found to be 56 feet 8 inches: determine the height of the flag-staff.

4. A stick 3 feet long is placed upright on the ground, and its shadow is found to be 4 feet 6 inches long: find the length of the shadow of a pole which is 45 feet high.

5. A country is 500 miles long: find the length of a map which represents the country on the scale of one-eighth of an inch to a mile.

6. The distance between two towns is 31 miles, and the distance between their places on a map is 7 inches: find the scale on which the map is drawn.

7. The distance between two towns is 54 miles, and the distance between their places on a map is 63 inches: find the distance between two other towns if the distance between their places on the map is 8 inches.

8. In the diagram of Art. 36 if BC=20 inches, DE=16, and BD=6, find BA.

9. In the diagram of Art. 36 if AD=8-inches, DE=7, and BD=3, find BC.

10. In the diagram of Art. 36 if DE=7 inches, BC=10, and BD=2, find DA.

11. The parallel sides of a trapezoid are respectively 16 and 20 feet, and the perpendicular distance between them is 5 feet; the other two sides are produced to meet: find the perpendicular distance of the point of intersection from the longer of the two parallel sides.

12. The parallel sides of a trapezoid are respectively 8 feet and 14 feet; two straight lines are drawn across the figure parallel to these so that the four are equidistant: find the lengths of the straight lines.

VII. CHORDS OF A CIRCLE.

78. Let AB be any chord of a circle, Cthe centre of the circle; suppose CD drawn perpendicular to AB, and produced to meet the circumference at E.

Then Dis the middle point of the chord ADB, and E is the middle point of the arc AEB. AB is the chord of the arc, and AE, or EB, is the chord of half the arc. DE is the height of the arc.

F

79. Produce EC to meet the circumference at F. The angle EAF is a right angle, by Art. 33. Hence, by Art. 37, the triangles EAF and EDA are similar, so that ED is to EA as EA is to EF. Therefore,

ED × EF EA × EA.

Also, by Art. 38,

ED × DF= AD × DB.

The present Chapter consists of applications of these two important results. We shall put the applications in the form of Rules for the sake of convenient reference, but any person who masters these two results will find it unnecessary to commit the Rules to memory.

80. Having given the height of an arc and the chord of half the arc, to find the diameter of the circle.

RULE. Divide the square of the chord of half the arc by the height of the arc, and the quotient will be the diameter of the circle.