(2) The two parallel sides of a trapezoid are 4.32 feet and 5'48 feet respectively; and the perpendicular distance between them is 2:18 feet.

Thus the area of the trapezoid is 10'682 square feet.

163. We have established the rule for finding the area of a trapezoid in a very simple manner in Art. 160; there is also another process which we will give as it is interesting and instructive.

Let ABCD be a quadrilateral having the sides AB and CD parallel. Through G the middle point of BC draw the straight line HGK parallel to AD, meeting the parallel sides of the trapezoid at H and K respectively.

Then the triangles BGH and CGK are equal; and thus the trapezoid ABCD is equivalent to the parallelogram AHKD. And since HB is equal to CK, it follows that AH is equal to half the sum of AB and CD. Thus the trapezoid is equivalent to a parallelogram having its base equal to half the sum of the parallel sides of the trapezoid, and its height equal to the perpendicular distance between those sides. Hence we have the rule given in Art. 161.

Through G draw a straight line parallel to AB meeting AD at L. Then L is the middle point of AD, and LG AH; so that half the sum of the parallel sides is equal to the straight line which joins the middle points of the other sides.

164. We will now solve some exercises.

(1) ABCD is a quadrilateral; B AB=3 feet, BC=4 feet,

CD=6 feet, DA=7 feet;

and the angle ABC is a right angle: find the area of the quadrilateral.

By Art. 55 we have AC equal to the square root of 9+16, that is to the square root of 25: so that AC=5.

The area of the triangle ABC=1×4×3=6.

D

The area of the triangle ACD can now be found by Art. 152.

1

5+6+7=18, of 18=9, 9—5—4, 9—6—3, 9—7—2,

2

9 × 4×3×2=216.

The square root of 216 cannot be found exactly; if we proceed to three decimal places we obtain 14.697. Thus the area of the quadrilateral is about 20'697 square feet.

(2) The diagonals of a rhombus are 80 and 60 feet respectively find the area; find also the length of a side, and the height of the rhombus.

× 80 × 60=2400. Thus the area is 2400 square feet.

The diagonals of a rhombus intersect at the middle point of each; thus to find the side of the rhombus we must determine the hypotenuse of a right-angled triangle the sides of which are 40 and 30 feet respectively. By Art. 55 the hypotenuse is the square root of 2500; so that the side of the rhombus is 50 feet.

2400

50

=48. Thus the height of the rhombus is 48 feet.

EXAMPLES. XIV.

Find the areas of the quadrilaterals having the following dimensions:

1. Diagonal 50'08 feet; perpendiculars 10-12 and 84 feet.

2. Diagonal 54 feet; perpendiculars 23 feet 9 inches and 18 feet 3 inches.

3. Diagonal 10 chains 14 links; perpendiculars 6 chains 27 links and 8 chains 6 links.

4. Diagonal 3 chains 27 links; perpendiculars 2 chains 15 links and 1 chain 75 links.

5. Diagonal 18 yards 2 feet, sum of the perpendiculars 16 yards 1 foot.

6. The area of a quadrilateral is 37 acres 1 rood 16 poles; one diagonal is 25 chains: find the sum of the perpendiculars on this diagonal from the two opposite angles.

Find the areas of the trapezoids which have the following dimensions :

7. Parallel sides 3 feet and 5 feet; perpendicular distance 10 feet.

8. Parallel sides 10 feet and 12 feet; perpendicular distance 4 feet.

9. Parallel sides 14 yards and 20 yards; perpendicular distance 12 yards.

10. Sum of the parallel sides 625 links; perpendicular distance 160 links.

11. Sum of the parallel sides 1225 links; perpendicular distance 240 links.

12. Parallel sides 750 links and 1225 links; perpendicular distance 1540 links.

4

13. The area of a trapezoid is 31 acres; the sum of the two parallel sides is 242 yards: find the perpendicular distance between them.

14. The area of a trapezoid is 8 acres 2 roods 17 poles; the sum of the parallel sides is 297 yards: find the perpendicular distance between them.

15. In Example 7 a straight line is drawn across the figure parallel to the parallel sides and midway between them: find the area of the two parts into which the trapezoid is divided.

16. In Example 9 two straight lines are drawn across the figure parallel to the parallel sides and dividing each of the other sides into three equal parts: find the areas of the three parts into which the trapezoid is divided.

17. The diagonals of a quadrilateral are 26 feet and 24 feet respectively, and they are at right angles: find the

area.

18. The diagonals of a rhombus are 88 yards and 110 yards respectively: find the area.

19. The diagonals of a rhombus are 64 yards and 36 yards respectively: find its area and the cost of turfing it at 4 pence per square yard.

20. The area of a rhombus is 52204 square feet, and one diagonal is 248 feet: find the other.

21. ABCD is a quadrilateral; AB=28 feet, BC=45 feet, CD-51 feet, DA=52 feet; the diagonal AC=53 feet: find the area.

22. ABCD is a quadrilateral; AB=48 chains, BC=20 chains, the diagonal AC=52 chains, and the perpendicular from D on AC=30 chains: find the area.

23. The sides of a quadrilateral taken in order are 27, 36, 30, and 25 feet respectively; and the angle contained by the first two sides is a right angle: find the

area.

24. The sides of a quadrilateral taken in order are 5, 5, 4, and 3 feet respectively; and the angle contained by the first two sides is 60°: find the area.

25. A railway platform has two of its opposite sides parallel and its other two sides equal; the parallel sides are 80 feet and 92 feet respectively; the equal sides are 10 feet each: find the area.

=

26. ABCD is a quadrilateral; AB 845 feet, BC=613 feet, CD=810 feet; AB is parallel to CD, and the angle at A is a right angle: find the area.

27. ABCD is a quadrilateral; the sides AB and DC are parallel. AB=165 feet,_CD=123 feet; the perpendicular distance of AB and DC is 100 feet. E is a point in AB such that AE is equal to half the difference of AB and CD: find the area of the triangle EBC, and of the quadrilateral AECD.

28. The diagonals of a rhombus are 88 and 234 feet respectively find the area; find also the length of a side, and the height of the rhombus.

29. The area of a rhombus is 354144 square feet, and one diagonal is 672 feet: find the other diagonal; find also the length of a side, and the height of the rhombus.

30. Two adjacent sides of a quadrilateral are 228 feet and 704 feet respectively, and the angle contained by them is 90°; the other two sides of the quadrilateral are equal, and the angle contained by them is 60°: shew that the area of the quadrilateral in square feet is

80256+136900/3.