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23. A hunter, wishing to determine his distance from a village in sight, took a point and from it laid off two lines in the direction of two steeples, which he supposed equally distant from him, and which he knew to be 100 rods asunder. At the distance of 50 feet on each line from the common point, he measured the distance between the lines, and found it to be 5 feet 8 inches. How far was he from the steeples?

5 ft. 8 in.: 100 rods:: 50 ft.: distance.

33 or, 68: 100 x X 2

×12::50: distance.

Ans.

14,559 feet, or nearly 3 miles.

24. A person is in front of a building which he knows to be 160 feet long, and he finds that it covers 10 minutes of a degree; that is, he finds that the two lines drawn from his eye to the extremities of the building include an angle of 10 minutes. What is his distance from the building?

Ans.

{

55,004 feet, or more than 10 miles.

REMARK. The questions of distance, with which we are at present occupied, depend for their solution on the properties of similar triangles. In the preceding example we apparently have but one triangle, but we have in fact two; the second being formed by the distances unity on the lines drawn from the eye of the observer, and the line which connects the extremities of these units of distance. This last line may be regarded as the chord of the are 10 minutes to the radius unity. We have seen that the length of the arc 180° to the radius 1, is 3.1415926; hence the chord of 1° or 60' is 0.017453, and of 10 it must be 0.0029089. Therefore, by similar triangles, we have

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25. In the triangle, ABC, we have given the angles A. = 32°, and B = 84°. The side AB is produced, and the exterior angle CBD thus formed, is bisected by the Cine BE, and the angle A is also bisected by the line AE, BE and AE meeting in the point E. What is the angle C, and what is the relation between the angles C and E?

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26. Suppose a line to be drawn in any direction between two parallels. Bisect the two interior angles thus formed on either side of the connecting line, and prove that the bisecting lines meet each other at right angles, and that they are the sides of a right-angled triangle of which the line connecting the parallels is the hypotenuse.

27. If the two diagonals of a trapezoid be drawn, show that two similar triangles will be formed, the parallel sides of the trapezoid being homologous sides of the triangles. What will be the relative areas of these triangles?

Ans.

The triangles will be to each other as the squares on the parallel sides of the trapezoid.

28. If from the extremities of the base of any triangle, lines be drawn to any point within the triangle, forming with the base another triangle; how will the vertical angle in this last triangle compare with that in the original triangle?

Ans.

It will be as much greater than the angle in the original triangle as the sum of angles at the base of the new triangle is less than the sum of those at the base of the first.

29. The two parallel sides of a trapezoid are 12 and 20, respectively, and their perpendicular distance is 8. If a line whose length is 14.5 be drawn between the inclined sides and parallel to the parallel sides, what is the area of the trapezoid, and what the area of each part, respectively, into which the trapezoid is divided?

Area of the whole, 128 square units;

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Dividing line at the distance of 2 from

shorter parallel side.

30. If we assume the diameter of the earth to be

7956 miles, and the eye of an observer be 40 feet above the level of the sea, how far distant will an object be, that is just visible on the earth's surface. (Employ Th. 18, B. III, after reducing miles to feet.)

Ans. 40992 feet = 7 miles 4032 feet.

31. The diameter of a circle is 4; what is the area of the inscribed equilateral triangle? Ans. 3√3.

32. Three brothers, whose residences are at the vertices of a triangular area, the sides of which are severally 10, 11, and 12 chains, wish to dig a well which shall be at the same distance from the residence of each. Determine the point for the well, and its distance from their residences.

REMARK. Construct a triangle, the sides of which are, respectively, 10, 11, and 12. The sides of this triangle will be the chords of a circle whose radius is the required distance. To find the center of this circle, bisect either two of the sides of the triangle by perpendiculars, and their intersection will be the center of the circle, and the location of the well.

Ans. The well is distant 6.405 chains, nearly, from each residence.

33. The base of an isosceles triangle is 12, and the equal sides are 20 each. What is the length of the perpendicular from the vertex to the base; and what the area of the triangle?

Ans. Perpendicular, 19.07; area, (19.07) x 6.

34. The hypotenuse of a right-angled triangle is 45 inches, and the difference between the two sides is 8.45 inches. Construct the triangle. Suppose the triangle drawn and represented by ABC, DC being the difference between the two sides.

Now, by inspection, we discover the steps to be taken for the construction of the triangle As AD = AB,

B

C

D

A

the angle ADB, must be equal to the angle DBA, and each equal to 45°.

Therefore, draw any line, AC, and from an assumed point in it as D, draw BD, making the angle ADB = 45°. Take from a scale of equal parts, 8.45 inches, and lay them off from D to C, and with Cas a center, and CB = 45 inches as a radius, describe an are cutting BD in B. Draw CB, and from B, draw BA at right angles to AC; then is ABC the triangle sought.

Ans. AB=27.3; AC = 35.76, when carefully constructed. 35. Taking the same triangle as in the last problem, if we draw a line bisecting the right angle, where will it meet the hypotenuse?

Ans. 19.5 from B; and 25.5 from C.

36. The diameters of the hind and fore wheels of a carriage, are 5 and 4 feet, respectively; and their centers are 6 feet asunder. At what distance from the fore wheels will the line, passing through their centers, meet the ground, which is supposed level? Ans. 24 feet.

37. If the hypotenuse of a right-angled triangle is 35, and the side of its inscribed square 12, what are its sides? Ans. 28 and 21.

38. What are the sides of a right-angled triangle having the least hypotenuse, in which if a square be inscribed, its side will be 12?

Ans.

The sides are equal to 24 each, and the least hypotenuse is double the diagonal of the square.

39. The radius of a circle is 25; what is the area of a sector of 50°?

REMARK. - First find the length of an arc of 50° in a circle whese radius is unity. Then 25 times that will be the length of an arc of the same number of degrees in a circle of which the radius is 25.

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3.14159265
180
1.04719755
6
25
x 125 x

× 5.

Area of sector =

1.04719755

=272.7077. Ans.

BOOK VI.

ON THE INTERSECTIONS OF PLANES, AND THE RELATIVE POSITIONS OF PLANES AND OF PLANES AND LINES.

DEFINITIONS.

A Plane has been already defined to be a surface, such that the straight line which joins any two of its points will lie entirely in that surface. (Def. 9, page 9.)

1. The Intersection or Common Section of two planes is the line in which they meet.

2. A Perpendicular to a Plane is a line which makes right angles with every line drawn in the plane through the point in which the perpendicular meets it; and, conversely, the plane is perpendicular to the line. The point in which the perpendicular meets the plane is called the foot of the perpendicular.

3. A Diedral Angle is the separation or divergence of two planes proceeding from a common line, and is measured by the angle included between two lines drawn one in each plane, perpendicular to their common section at the same point.

The common section of the two planes is called the edge of the angle, and the planes are its faces.

4. Two Planes are perpendicular to each other, when their diedral angle is a right angle.

5. A Straight Line is parallel to a plane, when it will not meet the plane, however far produced.

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