364. A TRIANGLE is a plain figure, which has three sides and three angles.

If a straight line meets another straight line, making the adjacent angles equal, each is called, a right angle; and the lines are said to be perpendicular to each other.

TRIANGLE is one

365. A RIGHT-ANGLED which has one right angle. In the right-angled triangle ABC, the side AC, opposite the right angle B, is called, the hypothenuse; the side AC, the base; and the side BC, the perpendicular.

366. We have seen (Art. 196), that the area of a square is equal to the product of two of its equal sides, which is the square of one side. Hence, the square root of the area of a square, will be the side itself.

Thus, if the area of the square in the figure

is 25, the square root of 25, which is 5, will Idenote the side.

367. In a right-angled triangle, the square described on the hypothenuse is equal to the sum of the squares described on the other two sides.

Thus, if ACB be a rightangled triangle, right-angled at C, then will the large square, D, described on the hypothenuse AB, be equal to the sum of the squares F and E, described on the sides AC and CB. This is called, the carpenter's theorem. By counting the small squares in the large square, D, you will find their number equal to that contained in the small squares F and E. In this triangle, the hypothenuse AB = 5, AC = 4,

and CB=3. Any numbers having the same ratio as 5, 4, and 3, such as 10, 8, and 6; 20, 16, and 12, &c., will represent the sides of a right-angled triangle.

1. Wishing to know the distance from A to the top of a tower, I measured the height of the tower, and found it to be 40 feet; also the distance from A to B, and found it 30 feet: what was the distance from A to C?

C

AC2 AB + CB2 =

2500

AC √2500 = 50 feet.

A

B

Hence, when the base and perpendicular are known, and the hypothenuse is required,

Square the base and square the perpendicular, add the results, and then extract the square root of their sum.

2. What is the length of a rafter that will reach from the eaves to the ridge-pole of a house, when the height of the roof is 15 feet, and the width of the building, 40 feet?

368. To find one side, when we know the hypothenuse and other side.

1. The length of a ladder which will reach from the mid

dle of a street, 80 feet wide, to the eaves of a house, is 50 feet what is the height of the house?

ANALYSIS. Since the square of the length of the ladder is equal to the sum of the squares of half the street and the height of the house, the square of the length of the ladder diminished by the square of half the street, will be equal to the square of the height of the house: Hence,

Square the hypothenuse and the known side, and take the difference; the square root of the difference will be the other side.

Examples.

1. If an acre of land be laid out in a square form, what will be the length of each side in rods?

2. What will be the length of the side of a square, in rods, that shall contain 100 acres?

3. A general has an army of 7225 men: how many must be put in each line, in order to place them in a square form?

4. Two persons start from the same point; one travels due east 50 miles, the other due south 84 miles: how far are they apart?

5. What is the length, in rods, of one side of a square that shall contain 12 acres?

6. A company of speculators bought a tract of land. for $6724, each agreeing to pay as many dollars as there were partners how many partners were there?

7. A farmer wishes to set out an orchard of 3844 trees, so that the number of rows shall be equal to the number of trees in each row: what will be the number of trees?

8. How many rods of fence will inclose a square field of 10 acres?

9. If a line 150 feet long, will reach from the top of a steeple 120 feet high, to the opposite side of the street, what is the width of the street?

10. What is the length of a brace whose ends are each 3 feet from the angle made by the post and beam?

CUBE ROOT.

369. The CUBE ROOT of a number is one of three equal factors of the number.

To extract the cube root of a number, is to find a factor which, multiplied into itself twice, will produce the given number.

Thus, 2 is the cube root of 8; for, 2×2×2 = 8: and 3 is the cube root of 27; for, 3×3×3 = 27.

The numbers in the first line are the cube roots of the corresponding numbers of the second.

370. A PERFECT CUBE is a number which has three exact factors. By examining the numbers in the two lines, we see,

1st, That the cube of units cannot give a higher order than hundreds.

2d, That since the cube of one ten (10) is 1000, and the cube of 9 tens (90), 81000, the cube of tens will not give a lower denomination than thousands, nor a higher denomination than hundreds of thousands.

Hence, if a number contains more than three figures, its cube root will contain more than one: if it contains more than six, its root will contain more than two, and so on; every additional three figures giving one additional figure in the root; and the figures which remain at the left hand, although less than three, will also give a figure in the root. This law explains the reason for pointing off into periods of three figures each.

371. Let us now see how the cube of any number, as 16, is formed. Sixteen is composed of 1 ten and 6 units, and may be written 10+6. To find the cube of 16, or of 10+6, we must multiply the number by itself twice.

1. By examining the parts of this number, it is seen that the first part, 1000, is the cube of the tens; that is,

10 x 10 x 10 = 1000.

2. The second part, 1800, is three times the square of the tens multiplied by the units; that is,

3 × (10)2 × 6 = 3 × 100 × 6 = 1800.

3. The third part, 1080, is three times the square of the units multiplied by the tens; that is,

3 × 62 × 10 3 x 36 x 10 = 1080.

4. The fourth part is the cube of the units; that is, 6 × 6 × 6 = 216,

63 =

1. What is the cube root of the number 4096?

ANALYSIS. Since the number

contains more than three figures, we know that the root will contain at least units and tens.

Separating the three righthand figures from the 4, we know that the cube of the tens

OPERATION.

4 096(16

1

12x33) 3 0 (9-8-7-6 163 4 096.

will be found in the 4; and 1 is the greatest cube in 4.

Hence, we place the root 1 on the right, and this is the tens of the required root. We then cube 1, and subtract the result. from the first period 4, and to the remainder we bring down the first figure, 0, of the next period.

We have seen that the second part of the cube of 16, viz., 1800,