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1. One side of a triangle may be regarded as its base. The perpendicular distance from its base (or from its base extended) to the opposite angle, is its altitude.

2. What is the altitude of the first of the above triangles? Of the second? Of the third ? 3. Convince yourself by measur

y ment and by paper cutting that every triangle is one half of a parallelogram having the same base and the same altitude as the triangle.

4. To find the area of a triangle, Find the area of the parallelogran: having the same base and altitude, and take one half of the result. Or, as the rule is given in the older books,Take one half of the product of the base and altitude."

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PROBLEM.—If the above figure represents a piece of land and is drawn on a scale of į inch to the rod, what part of an acre of land does it represent ?

285. MISCELLANEOUS REVIEW.

1. The specific gravity of granite is 2.7.* How much does a cubic foot of granite weigh?

2. A certain vessel is exactly large enough to contain 1000 grains of water. It will contain only 700 grains of petroleum. What is the specific gravity of the petroleum ? †

3. The specific gravity of gold is 19.3. How much does a cubic foot of gold weigh?

4. A cubic foot of sulphur weighs 125 lbs. What is the specific gravity of sulphur?

5. A cubic foot of steel weighs 487.5 lbs. What is the specific gravity of steel?

6. What is the ratio of .01 to .001 ? Of .1 to .01 ? Of 1 to .1?

7. What is the ratio of 1 bu. to 1 pk.? Of 1 pk. to 1 qt.? 8. What is the ratio of $371 to $15? Of $15 to $371 ?

9. What is the area of a rhomboid whose base is 16 inches and whose altitude is 16 inches?

10. Is the rhomboid described in problem 9 equilateral ?

11. The ratio of the perimeter of one square to the perimeter of another square is 4. What is the ratio of the areas of the two squares ?

12. Draw three triangles, the base of each being 4 inches and the altitude of each being 2 inches. Make one of them a right-triangle; another an equilateral triangle, and the third having angles unlike either of the other two. What can you say of the area of the right-triangle as compared with each of the others ?

* See page 185, Exercise 10. + This means what is the ratio of the weight of the petroleum to the weight of the same bulk of water ?

PROPORTION.

286. The terms of a ratio are together called a couplet. Two couplets whose ratios are equal are called a proportion.

The two couplets of a proportion are often written thus: 6:18 = 10:30, and should be read, the ratio of 6 to 18 equals the ratio of 10 to 30.

Couplets are sometimes written thus: 20:4 :: 50 : 10, and read, 20 is to 4 as 50 is to 10.*

287. TO FIND A MISSING TERM IN A PROPORTION.

EXAMPLE I.

36 : 12 :: x: 25.

The ratio of the first couplet is 3; that is, the antecedent is 3 times the consequent. Siuce the ratios of the couplets are equal, the ratio of the second couplet must be 3, and its antecedent must be 3 times its consequent. Three times 25 75, the missing term.

PROBLEMS.

Find the missing term.

1. 90 : 45 :: x: 180
2. 48:12 :: x : 150
3. 75 : 30 :: x : 140

4. 20 : 60 225
5. 30 : 50

= x : 175
6. 90 : 20

= x : 140

* The ratio sign (:) may be regarded as the sign of division (;-) with the hori. zontal line omitted, and the proportion sign (::) the sign of equality (=) with an erasure through its center, thus: (= =).

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Since the ratio of the first couplet is 3, the ratio of the second couplet must be 3, and x must equal 1 third of 48. 1 third of 48 is 16.

PROBLEMS.

Find the missing term.

1. 84 : 21
2. 96 :16 ::
3. 75 : 30 = 125 : x

172 : x 45 : x

4. 20: 60 :: 120 :
5. 25 : 35
6. 50: 25 :: 14 :x

45 : x

36 : x =

EXAMPLE III.

45 : 15. The ratio of each couplet is 3; so each consequent must be 1 third its antecedent, and x, 1 third of 36, or 12.

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The ratio of each couplet is 4; so each antecedent must be 4 times its consequent, and x, 4 times 12, or 48.

PROBLEMS.

Find the missing term.

1. x :16 ::51:17
2.

76 : 19
3. x:11: :24: 3

x : 22

4. x: 96 = 23 : 92
5. x: 40 :: 36 : 48

27 42:14

6. x :

* Since the ratio is 2/2 () the consequent must be of the antecedent.

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1. If 75 yd, of cloth cost $115.25, how much will 15 yd. cost at the same rate ?

75 yd. : 15 yd. $115.25 : x

2. If 23 acres of land cost $76.20, how much will 15 acres cost at the same rate ?

3. If 7 tons of coal can be bought for $26, how many tons can be bought for $39 ?

7 tons : x tons : : $26 : $39.

4. If 36 lb. coffee can be bought for $7, how many pounds can be bought for $174 ?

5. If sugar sells at the rate of 18 lb. for $1, how much should 63 lb. of sugar cost ?

18 lb.: 63 lb. = $1 : x dollars.

6. If a post 6 ft. high casts a shadow 4 feet long, how high is that telegraph pole which at the same time and place casts a shadow 20 feet long ?

7. If a post 5 feet high casts a shadow 8 feet long, how high is that steeple which casts a shadow 152 feet long?

8. If a train moves 50 miles in 1 hr. 20 min., at the same rate how far would it move in 2 hours ?

9. If a boy riding a bicycle at a uniform rate goes 12 miles in 1 hr. 15 min., how far does he travel in 25 minutes ?

TO THE TEACHER –After the pupil has solved the above problems by making use of the fact of the equality of the ratios, he should solve them by an analysis somewhat as follows: Prob. 1. Since 75 yd. cost $115.25, 1 yd. costs 7's of $115.25; but 15 yd. cost 15 times as much as 1 yd., so 15 yd. cost 15 times to of $115.25.

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