5. The result reached in the preceding article may be demonstrated more generally as follows:

Let ABC, A,B,C, be similar triangles, and let them be placed so that

contain n units, and

A,B, contain n,

units. Suppose

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1

A,B, divided into its units, and through the points of division draw lines parallel to BC or B,C1. Evidently the divisions of A,C, are all equal to one another, though not necessarily equal to those of A,B1. Then also AC contains n parts equal to AE, as AB contains n parts equal to AD; and A1C1 contains n1 parts equal to AE, as A,B1 contains n1 parts equal to AD. Hence

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In like manner the proportionality of the sides about the other equal angles may be shown.

6. On the other hand, if the lengths of the sides of one triangle may be obtained from the lengths of the sides of another by multiplying or dividing each by the same number; that is, if the sides of two triangles, taken in order, are proportional, what relation exists between the angles of the two triangles?

Construct and examine the following triangles, and see if you can supply an answer to the question: (1) Sides 20, 30, 40, and 40, 60, 80 millimetres. (2) Sides 1, 14, 12, and 14, 24, 2ğ inches. (3) Sides 24, 36, 40, and 42, 63, 70 millimetres.

Exercises.

1. The sides of two triangles are 20, 30, 40, and 40, 60, 80 millimetres, respectively. Construct them, and, using the bevel, show that they are equiangular.

2. The sides of two triangles are 20, 30, 40 and 30, 45, 60 millimetres, respectively. Construct them, and show that they are equiangular.

3. The bases of two triangles are 35 and 60 millimetres, and the angles adjacent to each base are 75° and 70°. Construct the triangles, and show that corresponding sides are as 35: 60.

4. Construct two triangles of different sizes with angles 35°, 45° and 100°. On a line AB lay off lines equal to the sides of one triangle ; and on another line AC lay off lines equal to the sides of the other triangle. Let the ends of corresponding lengths on AB, AC be joined. What position do these joining lines occupy with respect to each other? Apply test. What is the inference?

5. The angles of two triangles are 60°, 75° and 45°. Construct the triangles, and, after the manner suggested in question 4, test the proportionality of the sides.

6. The angles of two triangles are 110°, 30° and 40°, and the sides opposite angle of 30° in each are 40 and 55 millimetres. Construct the triangles, and, after the manner suggested in question 4, test the proportionality of the sides.

7. The angles at the vertices of two triangles are both 36°. The sides adjacent to the vertex of one triangle are 12 in. and 2 in., and adjacent to the vertex of the other 2ğ in. and 3 in. Construct the triangles. Show by measurement that angles opposite corresponding sides are equal, and that the remaining sides are in ratio 1 : 13.

8. The angles at the vertices of two triangles are both 67°, and the sides about these angles are 40, 60 and 44, 66 millimetres. Construct the triangles. Show by measurement that triangles are equiangular, and that the remaining sides are as 10: 11.

9. Construct an angle BAC of 39°, and from P in AC draw PN perpendicular to AB. Measure the lengths of AP, AN, PN in millimetres, and find the numerical values to two places of decimals of the ratios

10. In the preceding question, keeping to the angle of 39°, take the point P in different positions on AC, drop the perpendicular PN, for each position of P repeat the measurements and calculate to two decimal places the values of the preceding ratios. Compare values with those already obtained.

11. Keeping to same angle 39°, take the point P in AB and drop PN perpendicular on AC. Again calculate these ratios.

State your conclusion as to the values of these ratios,-perp. to hyp.; base to hyp.; perp. to base—so far as the angle 39° is concerned.

12. BC of a right-angled triangle ABC (C=90°) is found to be 748 ft., and the angle ABC is 39°. Use the results of the three preceding questions to find approximately the lengths of AC and AB in feet.

Similar Triangles. (Continued).

1. In the annexed figure the triangles ABC, ADE are similar. Suppose the values of the lines are

=

AD 59, AB=32, BC=24, and that DE is unknown. The property of similar triangles gives

2. If level ground can be found extending out from the base of a tree, or other vertical object, its height may be found as follows:

Let two rods, AB and CD, be placed upright in the ground, at such distance apart that the eye sees the tops (B and D) of the rods and the top (F) of the tree in the same straight line.

The heights of the rods being measured, their difference DG is known. Let also the lengths AC (i.e., BG) and CE (i.e., GH) be measured.

Then height of object, EF=34,111 + 13 = 47121⁄21⁄2.

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3. Suppose we wish to find the distance of an object B from A, without going over the distance AB with a surveyor's chain or other instrument for measuring.

Measure a base line, AC, of, say, 250 feet, and note the angles CAB, ACB. Then, on paper, construct a triangle A,B,C,, equiangular to ABC, but with a base line A,C, of, say, 1 foot. Measure the length, in feet, of A,B,. The line. AB will be 250 times the length of A1B1.

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This example embodies the principle of the range-finder, so much used in military and naval operations.

B