82. If from any point within an equilateral triangle, perpendiculars be drawn to the sides, the sum of these perpendiculars is equal to the perpendicular drawn from either of the angles to the opposite side. (Euclid, vi. 1.)

83. To find two straight lines which shall be arithmetic means between two given straight lines. (Deduction 9.)

84. To divide a given straight line AB by two points of division C and D, so that AC, AD, AB may be in harmonical proportion. (Euclid, vi. 4. Cape, iii. 69.)

85. If through any point D of a straight line AD which bisects a given angle BAC, a straight line GFDE be drawn, meeting AG (which is drawn perpendicular to AD), AB and AC respectively in G, F, E; show that ED, EF, EG are in harmonical proportion. (Euclid, vi. 4. Cape, iii. 69.)

86. If the four sides of a quadrilateral figure be bisected, the lines joining the points of bisection shall form a parallelogram, whose area equals half the area of the quadrilateral. (Euclid, vi. 2, I. Cape, iii. 66.)

87. If two diagonals of a regular pentagon intersect; 1st, the greater segment is equal to a side of the pentagon; 2nd, the two diagonals cut each other in extreme and mean ratio. (Euclid, vi. 4. Cape, iii. 69.)

88. Within an isosceles triangle to find a point, such that its distance from one of the equal angles may equal twice its distance from the vertical angle. (Euclid, iii. 31, and vi. 4. Cape, ii. 47, and iii. 69.)

89. If two circles touch each other, and any two parallel diameters be drawn, the straight line joining their extremities towards the same or opposite parts, according as the circles touch internally or externally, shall pass through the point of contact. (Euclid, vi. 4, and v. 17, 18. Cape, iii. 69, and Proportion, art. 180.)

90. If two circles touch each other externally, and also a given straight line, the part of the line between the points of contact is a mean proportional between the diameters. (Euclid, vi. 17. Cape, Proportion, art. 186.)

91. If two circles touch each other, either internally or externally, any two straight lines drawn from the point of contact will be cut proportionally by the circumferences. (Euclid, vi. 4. Cape, iii. 69.)

92. If from one extremity of a chord a tangent be drawn to a circle, equal to the chord, and a line be drawn joining the further extremities of the chord and the tangent, the arc intercepted between that line and the tangent shall be equal to half the arc subtended by the chord. (Euclid, vi. 33.)

93. If a straight line, which touches two circles, cut another

straight line, which joins their centres, the segments of the latter will be proportional to the diameters. (Euclid, vi. 4. Cape, iii. 69.)

94. If from the extremities of any chord of a circle, perpendiculars be drawn to the chord, the points where they meet any diameter shall be equally distant from the centre. (Euclid, vi. 2. Cape, iii. 66.)

95. If a circle be inscribed in a triangle, and another circle be described touching the base and the other two sides produced; Ist, the points where the circles touch the base shall be equally distant from its extremities; 2nd, the distance between the points where they touch either one of the sides shall be equal to the base. (Euclid, vi. 4, and v. 22. Cape, iii. 69, and Proportion, art. 183.)

96. To describe a circle which shall pass through two given points and touch a given straight line. (Euclid, vi. 13. Cape, iii. 72.)

97. From a given point in the side of a triangle, to draw a straight line, which shall bisect the triangle. (Euclid, i. 37. Cape, iii. 53. Cor. 2.)

98. From a given angle of a trapezium, to draw a straight line, bisecting the trapezium. (Euclid, i. 37. Cape, iii. 53. Cor. 2.)

99. From a given point in the side of a triangle, to draw straight lines, which shall divide it into any number of equal parts. (Euclid, i. 37, and vi. I. Cape, iii. 53. Cor. 2.)

100. To transform any rectilineal figure into a triangle, of equal area, whose vertex shall be in one of the angles of the figure, and its base in one of its sides. (Euclid, i. 37. Cape, iii. 53. Cor. 2.) IOI. To transform any given triangle into an isosceles one of equal area. (Euclid, i. 37. Cape, iii. 53. Cor. 2.)

102. To transform any given isosceles triangle into an equilateral one of equal area. (Euclid, vi. 13. Cape, prob. 9.)

103. To divide a given straight line into two segments, such that the rectangle contained by them shall be a maximum. (Euclid, i. 10. Cape, prob. 1.)

104. Through a given point within a circle, to draw the least possible chord. (Euclid, iii. 7, 15. Cape, ii. 37.)

105. On a given base, to describe a triangle, having a given vertical angle, and whose area shall be a maximum. (Euclid, iii. 33, 15. Cape, prob. 22, and ii. 37.)

106. On a given base, to describe a triangle, another of whose sides is given, so that the area may be a maximum. (Euclid, i. 11. Cape, prob. 2.)

107. To divide a circle into any number of parts, which shall be equal both in area and in perimeter. (Euclid, v. 17, and xii. 2. Cape, iv. 87, and Proportion, art. 180.)

18. To divide a circle into any number of equal concentric an

nuli. (Euclid, vi. 9, 8, xii. 2, and v. 17. Cape, iii. 72, iv. 87, and Proportion, art. 180.)

109. Given one side of a right-angled triangle, and the difference between the hypothenuse and the other side, to construct it. (Euclid, vi. 12, and i. 47. Cape, prob. 8, and iii. 61.)

110. Given the perpendicular from the right angle on the hypothenuse of a right-angled triangle, and the difference of the segments of the hypothenuse, to construct it. (Euclid, i. 47. Cape, iii. 61.)

III. Given the hypothenuse, and the sum of the two sides of a right-angled triangle, to construct it. (Euclid, i. 32, iii. 20, 21. Cape, i. 24, ii. 44, 45.)

112. Given the segments of the hypothenuse of a right-angled triangle, made by a perpendicular from the right angle, to construct it. (Euclid, iii. 31. Cape, ii. 47.)

113. Given the hypothenuse, and the difference of the sides of a right-angled triangle, to construct it. (Euclid, iii. 31, 26; i. 6. Cape, ii. 47; i. 9.)

114. Given the sides of a right-angled triangle in continued proportion, and the length of the hypothenuse, to construct it. (Euclid, ii. 11; vi. 8. Cape, prob. 10, and iii. 72.)

115. Given the base of a triangle, one of the angles at the base, and the difference of the two sides, to construct it. (Euclid, i. 6. Cape, i. 9.)

116. Given the base, the difference of the two angles at the base, and the difference of the two sides of a triangle, to construct it. (Euclid, i. 23, 6, 32. Cape, prob. 4, and i. 9, 24.)

117. Given the vertical angle of a triangle, and the segments into which the perpendicular from the vertex divides the base, to construct it. (Euclid, iii. 33. Cape, prob. 22.)

118. Given the base, the vertical angle, and the sum of the sides of a triangle, to construct it. (Euclid, iii. 33, 20. Cape, prob. 22, and ii. 44.)

119. Given the base, the ratio of the sides, and the vertical angle of a triangle, to construct it. (Euclid, iii. 33, vi. 10. Cape, prob. 22.)

120. Given the vertical angle of a triangle, the sum of its sides, and the difference of the segments into which a perpendicular from the vertex divides the base, to construct it. (Euclid, i. 23, 32; iii. 3. Cape, prob. 4; i. 24; ii. 35.)

74

MENSURATION.

AREAS OF PLANE FIGURES.

General Formulæ.

1. In a triangle, if a, b, c be the sides opposite to the angles A, B, C; d the perpendicular from A on a, p the semi-perimeter,

area

2. In a

angle A,

12

ad = bc sin = √p(p-a)(p—b)(p—c).

2

parallelogram, if b, c be two sides including the

area bc sin A.

3. In a trapezoid, if a, b be the two parallel sides, d the perpendicular distance between them,

4. In a regular polygon of n sides, if za be the length of one of the sides,

6. In a circular ring, if a and b be the external and internal radii,

area= = π(α2 — b2).

7. In the sector

at the centre,

length of arc

area of sector

of a circle, if n be the number of degrees

whole circumference of circle :: n: 360, area of circle ::n: 360.

8. In a parabola, if a be the height, and 6 the base,

1. Find the area of a square, whose side is 15 chains, 40 links. 2. Find the area of a rectangular field, whose dimensions are 503 yards, and 123 yards.

3. Find the area of a rhombus, whose side is 5 feet 7 inches, and perpendicular height 4 feet.

4. Find the area of a rhombus, whose side is 17 yards, and one of whose angles is 49° 14' 15".

5. Find the area of the parallelogram, two adjacent sides of

which are 4 chains, and 5 chains 50 links respectively; the angle included between them being 16° 43'.

6. A triangular field 738 links long, and 583 links in the perpendicular, produces an income of £12 a-year. At how much an acre is it let?

7. How much ground is there in a triangular fish-pond, whose three sides measure 400, 348, and 312 yards respectively?

8. What is the area of a triangle, two adjacent sides of which are 24 and 17.6 yards, and include an angle of 30°?

9. Having a rectangular marble slab, 36 inches by 16 inches, I would have a square foot cut off it, parallel to the shorter side: I would then have the like quantity taken from the remainder, parallel to the longer side, and so on, as often as possible. What then would be the remainder?

10. I have to plant 10584 trees, at equal distances in rows, in a plantation whose length is six times its breadth. How many rows will there be in the shorter end?

11. If I place 3582 plants in rows, each 4 feet asunder, and the plants 7 feet apart; how much ground is taken up?

12. What is the area of the bottom of a bath, the sum of whose three equal sides is 125 feet?

13. How much paper yard wide will be required for a room, that is 22 feet long, by 14 feet wide, and 9 feet high; if there be 3 windows and 2 doors, each 6 feet by 3 feet?

14. How many square feet are there in a plank, whose length is 10 ft. 5 in., and the breadths of the two ends 2 feet, and 13 feet respectively?

15. How many square yards of paving are there in a quadrangular court, whose diagonal is 54 feet; and the perpendiculars on it from the opposite corners 25 and 173 feet?

15

feet.

16. Find the area of a regular pentagon, whose side is 17. Find the area of a regular hexagon, whose side is 15 feet. 18. Find the area of a regular heptagon, whose side is 15 feet. 19. Find the area of a regular octagon, whose side is 15 feet. 20. If the diameter of a circle be 5 feet, what is its circumference? 21. If the circumference of a circle be 10 chains, what is its radius?

22. What is the area of a circle, whose diameter is 12 feet? 23. How many square feet are there in a circle, whose circumference is 6.2832 feet?

24. What is the length of an arc of a circle containing 2910; the radius of the circle being 9 feet?

25. Find the area of the sector of a circle, whose radius is 55 yards; the length of the arc being 59 yards.

26. If the diameter of a circle be 84 inches, what is the area of a segment whose height is 30 inches?