5. The base and perpendicular of a triangle are 120 and 48, and the perpendicular from one of the angles at the base on the opposite side is 72; find the other two sides. Ans. 80 and 73'756.

6. Find the area of an equilateral triangle whose side is 56 yards. Ans. 443 P.

THE TRAPEZIUM.

VI.

To find the area of a trapezium when the diagonal and the perpendiculars upon it from the opposite angles are given.

Multiply the sum of the perpendiculars by the diagonal, and divide by 2.

D

E

Ex. I.

Let ABCD be a trapezium, of which the diagonal is AC, and BE, DF perpendiculars upon AC from the opposite angles B, D. Let AC-324 yards, BE = 194 yards, and DF= 245 yards.

245

194

439

324

1756

878

1317

2 142236

4840 71118 (14a. 2 r. 31 p.

4840

22718

19360

1210) 3358

2420

938

40

3752

363

122

121

I

Ex. 2.

Find the value of a board in the form of a trapezium, whose diagonal is 5 ft. 9 in., and perpendiculars upon it from the opposite angles, 4 ft. 6 in., and 6 ft. 8 in., at is. 9d. per square yard.

I. The diagonal of a trapezium is 18 ft. 4 in., and the perpendiculars upon it from the opposite angles are 10 ft. 6 in., and 12 ft. 8 in. Find the area. Ans. 212 sq. ft.

2. The diagonal of a trapezium is 93 chains, and the perpendiculars upon it are 6 and 103 chains; what will it cost mowing at 8s. 6d. per acre?

Ans. £3. 8s. 11d. nearly.

3. The area of a trapezium is 16 a. 3 r. 8 p., the diagonal 16 chains, and the perpendiculars upon it are in the ratio of 5 to 7; find the perpendiculars.

4.

Ans. 8, 12 chains. 121

ABCDE is a five-sided figure; the diagonal AD is 7 chns. 15 links, and the perpendiculars from C and E upon it are 4 chns. 12 lks., and 5 chns. 62 lks.; AB is 5 chns. 57 lks., BC 1 chn. 65 lks., and the angle ACB is a right angle. Find the area. Ans. 3 a. 3 r. 27,445 P.

Let ABCD be a trapezoid, of which the parallel sides are AB, CD; and let DE be perpendicular to AB. Then DE is the breadth of the trapezoid.

VII.

To find the area of a trapezoid.

Multiply the sum of the parallel sides by the breadth, and divide by 2.

Ex. Find the value of a piece of ground in the form of a trapezoid, whose parallel sides are 28 ft. 6 in. and 22 ft. 3 in., and the breadth 13 ft. 6 in., at 16d. per yard.

EXAMPLES.

I. Find the area of a trapezoid whose parallel sides are 3 ft. 7 in., and 5 ft. 6 in., and the breadth 6 ft. 5 in. Ans. 29.7.2 6 sq. ft.

2. Find the area of a trapezoid whose parallel sides are 131 and 243 yards, and the breadth 352 yards.

Ans. 13 a. 2 r. 16 p.

3. The parallel sides of a field in the form of a trapezoid are 8 chns. 15 lks., and 10 chns. 45 lks., and the breadth 6 chns. 24 lks.; find the area, and the rent of the field for 9 months at 2 guineas per acre.

4.

Area, 5 a. 3 r. 81 p.

Rent, £9. 25. 93d.

Find the expense of digging a garden in the form of a trapezoid, whose parallel sides are 82 and 94 yards, and the breadth 33 yards, at 3d. per statute perch.

Take any straight line AB, and at the points C, D, E, &c., let Cc, Dd, Ee, &c., be drawn at right angles to AB, and join Ac, cd, de, &c. The lines Cc, Dd, Ee, &c. are called offsets. Each of the portions ACc, CDdc, &c., may be measured by the rules for the triangle and trapezoid.