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(18) The length of a rectangular field is 148 links, and its breadth 111 links, find the distance from corner to corner.

(19) A room is 30 feet long and 16 feet broad; what length of line would stretch between its opposite corners?

(20) A footpath runs along the two adjacent sides of a rectangular common whose area is 60 square chains, and breadth 5 chains. Find the distance saved by going straight from one corner to the opposite corner across the field, instead of keeping to the footpath.

72. To find one side of a right-angled triangle when the hypotenuse and the other side are given.

RULE. From the square of the hypotenuse subtract the square of the given side, and extract the square

root of the remainder.

See Fig. Art. 71.

Sq. on A C+sq. on B C=sq. on A B; or, 16+9=25. .. sq. on A C=sq. on A B-sq. on B C; or, 16=25-9. And, sq. on B C=sq. on A B-sq. on A C; or, 9-25-16. Ex. The hypotenuse is 15 ft., and one side is 12 ft.; find the other.

Square of hypotenuse =225.

Square of given side =144.

Square of side required=225-144-81.

Length of side required sq. root of 81-9.

Ex. XXVIII.

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

(1) 61 yds., 11 yds.
(2) 101 ft., 20 ft.
(3) 37 lks., 35 lks.

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(4) 67.7 chs., 5.2 chs.
(5) 39 yds., 5 yds.
(6) 1,34.

(7) If the hypotenuse is 2501, and the perpendicular 100, what is the base?

(8) The base of a right-angled triangle is 3-12, and the hypotenuse 3.13; find the perpendicular.

(9) A ladder 51 feet long reaches a window 45 feet from the ground; find the distance of the foot of the ladder from the side of the house.

(10) A ladder 34 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 four feet.

[When the foot of the ladder is pulled out, it reaches to the height of thirty feet.]

(11) A line of 117 yards will reach from the top of a castle 108 yards high, standing at the side of a river, to the opposite bank; required the breadth of the river.

E

(12) A ladder 65 feet long, being placed in a street, will exactly reach to a window 63 feet from the ground on one side, and upon being turned over without moving the foot, will reach a window 39 feet high on the other side; required the breadth of the street.

[First find A B; then PC; the sum of the two will be the breadth of the street.]

House.

A

Ladder.

Ladder.

D

House.

(13) The bottom of a ladder is placed at a point in a street 15 feet from a house, and the top of the ladder rests against the house at 20 feet from the ground; and on turning the ladder over to the other side of the street, its top rests at 24 feet from the ground; find the breadth of the street.

B

Street.

(14) A ship's mast, partially broken off by a blast of wind, strikes the deck with its top at a distance of 10 feet from the foot of the mast; find the height of the whole mast, supposing the broken piece to be 26 feet long.

[The upper portion falling to the deck is the hypotenuse of a rightangled triangle, and its length is known. The base is also known, hence we are to determine the perpendicular, or height of the part of the mast still standing, from which we readily obtain the entire length of the mast.]

(15) A may-pole, snapped off in a storm, strikes the ground at a distance of 16 ft. from its foot. The portion still upright is 30 ft.; find the whole length.

(16) In a gale a flagstaff 25 yards high, standing on level ground, snaps 12 yards from the bottom, and the upper portion not being wholly broken off, the top touches the ground. How far is its point of contact with the ground from the bottom of the staff?

(17) The length AB of a rectangle ABCD is 720, and its diagonal A C is 724, find the breadth B C.

[See Ex. XXVII., Quest. 17.]

(18) The breadth of a rectangular field is 23 yds., and its diagonal 265 yds.; find the area.

(19) Two travellers, A and B, arrive at the corner of a rectangular lake. A goes to the opposite corner in a boat, a distance

of 1 miles; B walks along the shore of the lake, intending to rejoin him, and has to go a mile before he turns the end of the lake; how much farther has B to go before he rejoins A?

Ex. XXIX.

Miscellaneous Examples on Parallelograms and Triangles.

(1) A square field contains 5 ac. 1 ro. 1 po.; find the length of its side in chains and links.

(2) Find the cost of carpeting a room 24 feet by 18 feet, with carpet 27 inches wide, at 5s. 4d. per yard.

(3) Each side of a rhombus is 65 feet, and one of the diagonals is 104 feet. Find the area in square feet.

(4) Find the area, in acres, etc., of a rectangle whose sides are 5 chains 14 links, and 6 chains 25 links.

(5) Find the area, in square yards, of a triangle whose sides are 130, 140, and 150 yards long.

(6) Find the hypotenuse of the right-angled triangle whose sides are 17 and 144.

(7) How many yards of paper, 27 inches wide, will be required to cover the four walls of a room 37 feet square and 10 feet 6 inches high?

(8) The top of a ladder touches a wall 4 yards above the ground, when its foot is placed 3 feet 6 inches from the wall. How long is the ladder?

(9) Find by duodecimals the area of a square whose side is 17 feet 8 inches, and express the result in sq. yds., ft., and in. (10) A courtyard whose area is 810 square feet is to be paved with 1080 rectangular tiles. The breadth of each tile is 9 inches, find the length of each.

(11) Find the area, in square yards, of a field, the sides of which are 270, 360, and 450 yards respectively.

(12) What is the length of a ladder which just reaches to the top of a wall 36 feet high when its foot is 48 feet from the wall? (13) Find the cost of papering a room 21 feet 9 inches long, 15 feet 7 inches broad, and 8 feet 11 inches high, with paper 22 inches broad, at 1s. 6d. per yard.

(14) The sides of a rectangle are 17 yds. and 425 yds.; find the side of a square of equal area.

(15) Find the number of turfs, 4 feet by 8 inches, required for a garden plot 17 feet by 72 feet.

(16) Find the diagonal of a rectangle whose adjacent sides are 168 and 26.

(17) Find the rent of a triangular field whose base is 12 chs. 85 lks., and perpendicular 2 chs. 50 lks., at £4 per acre.

(18) A church and chapel stand opposite each D other in a street 24 yds. wide; the height of the highest point of each building, rising perpendicularly from the side of the street, is 120 feet and 75 feet respectively. Find the distance from the top E of the one to the top of the other.

[Let AD represent the height of the church, BC the height of the chapel, and A B the width of the street. Then DĚ= 120 75 45, and E C=24.]

(19) Find the area of a triangle whose sides are 55, 65, and 100.

(20) Determine the side of a square field which cost £115 10s. 5d. trenching, at 5 d. per sq. yard.

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(21) A room 52 ft. long requires 48 yds. of carpet, 26 in. wide, to cover it; what is the breadth of the room?

(22) What is the cost of papering a room 30 feet long, 20 feet wide, and 21 feet high, with paper which costs 9d. a yard, and is 2 feet 4 inches wide?

(23) A city footway 12 feet wide requires a quarter of a million of tiles, 6 inches square, to lay it down; what is the length of the footway in yards, etc.?

(24) The height of a tower on a river's bank is 165 feet; the length of a line from the top to the opposite bank is 557 feet; what is the breadth of the river?

(25) A cottage which stands on a D square plot of 729 feet, contains four rooms of equal size on the groundfloor, and is divided into two equal parts by a passage four feet wide, having two rooms on each side; allowing one foot for the thickness of each wall, find the area of each of the four rooms.

[Side of plot-sq. root of 729=27 ft.] From AD take the sum of four times the thickness of each wall and the width of the passage, and one-half the remainder will give one dimension of each room.

A

C

B

From DC take three times the thickness of each wall, and onehalf the remainder will give the other dimension.]

(26) How many yards of velvet trimming, yard wide, can be cut from 11 yard of velvet, which is yard wide? and what is the whole number of yards of trimming worth at 24d. per yard? (27) A rectangular piece of ground of 15 acres, 3 roods, 3 poles, is one-third as broad as it is long. What is the distance round it, in chains?

(28) Find the area in sq. yds., of a field the sides of which are 60, 52, and 56 yards respectively.

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