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Thus the sum of the hypotenuse and the other side is 578, and their difference is 288. Add and divide by 2; thus we obtain 433, which is the hypotenuse. Subtract 433 from 578 and we obtain 145, which is the other side.

(2) Each side of an equilateral triangle is 1 foot required the height of the triangle.

Let ABC be the triangle; CD the height. CD will divide AB into two equal parts;

thus AD=. We shall find 2

CD by the second form of the rule in Art. 60,

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D B

1

X =

The square root of

3 1
4

=

2

of the square root of 3. The

square root of 3 cannot be found exactly; if we proceed to three decimal places we obtain 1.732, and half of this is 866. Thus the height is 866 feet approximately.

=

(3) The base of a triangle is 56 feet, the height is 15 feet, and one side is 25 feet: required the other side. Let AB=56, CD: 15, BC=25. We first find BD by Art. 60. 25+ 15 = 40, 25 15 10, 40 × 10-400: the square root of 400 is 20. Thus DB=20.

=

Therefore AD=56-20=36.

A

B

Then we find AC by Art. 55. The square of 36 is 1296, and the square of 15 is 225; the sum of 1296 and 225 is 1521: the square root of 1521 is 39. Thus AC=39 feet.

EXAMPLES. V.

Determine the hypotenuse from the given sides in the following right-angled triangles:

1. 532 feet, 165 feet.

2. 7584 feet, 3937 feet.

3. 278 feet 8 inches, 262 feet 6 inches.

4. Half a mile, 345 yards 1 foot.

Determine in feet, as far as two decimal places, the hypotenuse from the given sides in the following right-angled triangles :

5. 437 feet, 342 feet.

6. 4395 feet, 3874 feet.

7. 314 feet 3 inches, 228 feet 9 inches.

8. A quarter of a mile, 427 yards 2 feet.

Determine the other side from the given hypotenuse and side in the following right-angled triangles :

9. 725 feet, 644 feet.

10.

11.

16417 feet, 14208 feet.

269 feet 5 inches, 250 feet 8 inches.

12. 340 yards 1 foot, one furlong.

Determine in feet, as far as two decimal places, the other side from the given hypotenuse and side, in the following right-angled triangles:

13. 647 feet, 431 feet.

14. 4987 feet, 3765 feet.

15. 424 feet 3 inches, 276 feet 6 inches.

16. 5 furlongs, 916 yards 2 feet.

17. The sides of a triangle are 22620 feet and 12815 feet, and the height is 11484 feet: find the base.

18. One side of a right-angled triangle is 3925 feet; the difference between the hypotenuse and the other side is 625 feet: find the hypotenuse and the other side.

19. A ladder 25 feet long stands upright against a wall: find how far the bottom of the ladder must be pulled out from the wall so as to lower the top one foot.

20. A ladder 40 feet long is placed so as to reach a window 24 feet high on one side of a street, and on turning the ladder over to the other side of the street it reaches a window 32 feet high: find the breadth of the street.

21. The bottom of a ladder is placed at a point 14 feet from a house, and the top of the ladder rests against the house at 48 feet from the ground; and on turning the ladder over to the other side of the street its top rests at 40 feet from the ground: find the breadth of the street.

22. Find to ten decimal places the diagonal of a square of which the side is one inch.

23. A side of a square is 110 feet: find the diagonal.

24. The radius of a circle is 82.66 feet; the perpendicular drawn from the centre on the chord is 711 feet: find the chord.

25. A footpath goes along two adjacent sides of a rectangle; one side is 196 yards, and the other is 147 yards find the saving in distance made by proceeding along the diagonal instead of the two sides.

26. The span of a roof is 28 feet; each of its slopes measures 17 feet from the ridge to the eaves: find the height of the ridge above the eaves.

27. The side of a square is 8 feet: find the radius of the circle described round the square.

28. The radius of a circle is 6 feet: find the side of a square inscribed in the circle.

29. The radius of a circle is 7 feet: find the perpendicular from the centre on a chord 8 feet long.

30. The radius of a circle is 17 inches; the perpendicular from the centre on a chord is 13 inches: find the chord.

31. The radius of a circle is divided into six equal parts, and at the five points of division straight lines are drawn at right angles to the radius to meet the circumference: find the lengths of these straight lines, in inches to three decimal places, that of the radius being one foot.

32. The radius of a circle is 7 feet; from a point at the distance of 12 feet from the centre a straight line is drawn to touch the circle: find the length of this straight line.

VI. SIMILAR FIGURES.

66. Let ABC and DEF be two similar triangles. Then AB is to BC as DE is to EF; see Art. 34.

A A

Thus if two sides of one triangle be given, and one of the corresponding sides of a similar triangle, the other corresponding side of the second triangle can be found. The process will be that which is known in Arithmetic as Proportion or the Rule of Three.

67. Examples:

(1) Suppose AB=5, BC=6, DE=7,

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(2) Suppose AB=5, AC=4, DE=7,

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68. Similar triangles frequently present themselves in the theory and practice of mathematics.

For example, we found in Art. 65 that if the side of an equilateral triangle be 1 foot the height is 866... feet. Now this proportion will always hold between the side and the height of an equilateral triangle; so that if the side of an equilateral triangle be 7 feet the height will be 7x866... feet.

Again, we have said that the triangles AED and BEC in the diagram of Art. 38 are similar; so that EA is to ED as EC is to EB. Hence it follows by the usual theory of Proportion that the product of EA into EB is equal to the product of EC into ED; this is a very remarkable and very important property of the circle.

69. By the aid of similar triangles we can determine the height of an object when we have measured the length of its shadow.

For example, suppose that a stick is fixed upright in the ground, and that the height of the portion above the ground is 3 feet and the length of the shadow 4 feet. Also suppose we find at the same time that the length of the shadow of a certain tree is 52 feet. Then we determine the height of the tree by the proportion

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70. From similar triangles we pass naturally to the consideration of similar rectilineal figures.

Similar rectilineal figures are those which have their several angles equal, each to each, and the sides about the equal angles proportionals.

71. Take for example two five-sided figures ABCDE and abcde; these are similar if the angles at A, B, C, D, E

E

B

are equal to the angles at a, b, c, d, e respectively, and the

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