522. To find any side of a right-angled triangle when the other sides are given.

523. A Triangle is a figure which has three angles and three sides.

524. A Right Angle is the angle formed when one line is drawn perpendicular to another.

525. A Right-angled Triangle is a triangle which has a right angle.

526. The Hypotenuse of a right-angled triangle is the side opposite the right angle.

527. The Base of a triangle is the side on which it is assumed to stand.

528. The Perpendicular is the side which forms a right-angle with the base.

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The relation of the squares described upon the sides of a rightangled triangle is expressed thus:

529. PRINCIPLES.-1. The square described upon the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides.

2. The square on either of the other sides of a right-angled triangle is equal to the square on the hypotenuse diminished by the square on the other side.

When the number of square units in the surface of any square figure is known, its side may be found by extracting the square root of the number according to the preceding case.

530. 1. The base of a right-angled triangle is 8 feet and the perpendicular 6 feet, what is the hypotenuse?

PROCESS. 8262100.

V100

= 10.

ANALYSIS.-Before we can determine the length of the hypotenuse when the sides are given we must find the area of a square described upon it. The square described upon it is equal to the sum of the squares upon the other two sides, or the sum of 82+62, which is 100. Since the area of the square described upon the hypotenuse contains 100 square units, the length of the side is the square root of 100 or 10.

To find the hypotenuse.

RULE.-Extract the square root of the sum of the squares of the other two sides.

To find the base or perpendicular.

RULE.-Extract the square root of the difference of the squares on the hypotenuse and the other side.

2. The base of a right-angled triangle is 15 feet and the perpendicular is 20 feet. What is the hypotenuse?

3. The base of a right-angled triangle is 40 feet and the hypotenuse is 120 feet. What is the perpendicular?

4. The perpendicular of a right-angled triangle is 30 feet and the hypotenuse is 50 feet. What is the base?

5. A tree 150 feet high, standing upon the bank of a stream, was broken off 125 feet from the top, and falling across the stream the top just reached the other shore. What was the width of the stream?

6. Two steamers start from the same point, one going due north at the rate of 15 miles an hour, and the other going due

west at the rate of 18 miles an hour. How far apart were they at the end of 6 hours?

7. A rectangular park, whose sides are respectively 45 rods and 60 rods in length has a walk crossing it from corner to corner. How long is the walk?

8. A certain assembly room is 100 feet in length, 60 feet in width, and 26 feet in height. What is the distance from a lower corner to the upper opposite corner?

9. Two buildings standing opposite each other are respectively 60 feet and 80 feet high. A ladder 125 feet long placed at a certain distance from the base of each just reaches the top of each. How far apart are the buildings?

10. The distance from the base of a building to a pole is 145 feet, and a string 225 feet long attached to the top of the pole just reaches the base of the building. What is the height of the pole?

11. A person who wished to ascertain the exact height of St. Paul's Cathedral in London, England, learned by inquiry that a rope extending from the top of the cross to a point 300 feet from the center of the circular pavement under the dome was 488 feet 10 inches long. If these data were correct, what is the height of St. Paul's?

SIMILAR FIGURES.

531. Similar Figures are such as are of the same form, and differ from each other only in size.

The truth of the following principles can be shown by geometry:

532. PRINCIPLES.-1. Similar surfaces are to each other as the squares of their corresponding dimensions. Hence,

2. The corresponding dimensions of similar surfaces are to each other as the square roots of their areas.

1. If the area of a triangle whose base is 16 rods, is 128 square rods, how many square rods are there in the area of a similar triangle whose base is 12 rods ?

PROCESS.

128 x 162: 122 or, 128 x 256: 144

ANALYSIS.-Since the areas of similar figures are to each other as the squares of their like dimensions, the area of the first triangle (128 sq. rd.) will be to the area of the second triangle (x) as the square of the side of the first triangle (162) is to the square of the side of the second triangle (122). Solving the proportion, the area is 72 sq. rd.

x= 72 sq. rd.

2. If the area of a circle, whose diameter is 2 feet, is 6.2832 sq. ft. what will be the area of a circle whose diameter is 4 feet?

3. If the side of a rectangular field containing 25 acres is 40 rods, what is the side of a similar field containing 10 acres? ANALYSIS. Since the corresponding dimensions of similar surfaces are to each other as the square roots of their areas:

√25: √10 :: 40: x, or 5: √10 :: 40: x.

Extracting the square root of 10 and solving the proportion, x, or the corresponding side, is 25.296 rd.

4. If the side of a square field containing 40 acres is 80 rods, what will be the side of a similar field whose area is 25 acres?

5. If the area of a circle whose diameter is 20 feet is 314.16 square feet, what is the diameter of a circle whose area is 113.0976 square feet?

6. A farmer has two rectangular fields similar in form: one, whose length is 120 rods and whose breadth is 12 rods, contains 9 acres, the other contains 6 acres. What are its length and breadth?

7. A horse tied to a stake by a rope 7.13 rods long can graze upon just 1 acre of ground. How long must the rope be that he may graze upon 5 acres?

CUBE ROOT.

533. 1. What is the cube root of 13824, or what is the edge of a cube whose solid contents are 13824 units?

figures each, beginning at units.

ANALYSIS.-According to Pr. 2, Art. 519, the orders of units in the cube root of any number may be determined from the number of periods obtained by separating the number into pe

riods, containing three

Separating the given number thus, there are two periods, or the root is composed of tens and units.

The tens in the cube root of the number can not be greater than 2, for the cube of 3 tens is 27000. 2 tens, or 20 cubed, are 8000, which, subtracted from 13824, leave 5824; therefore the root, 20, must be increased by a number such that the additions will exhaust the remainder.

The cube (A) already formed from the 13824 cubic units is one whose edge is 20 units. The additions to be made, keeping the figure formed a perfect cube, are 3 equal rectangular solids, B, C and D; 3 other equal rectangular solids, E, F and G; and a small cube, H. Inasmuch as the solids, B, C and D, com