PART II.

MENSURATION OF LINES AND SURFACES.

LESSON IV.-RIGHT-ANGLED TRIANGLES.

A TRIANGLE is said to be a rectilinear figure, i.e., made up of right lines or straight lines. It is called a triangle because it has three angles, or (which is exactly the same thing) three sides.

In the same way four-sided figures, of which we shall speak a little farther on, are called quadrangles.

Now there is a great deal which is very important and interesting about every triangle, as you have either already begun to learn, or will have to learn shortly when you commence Euclid and Trigonometry. You can never know very much of mensuration or of land-surveying (which is only one application of mensuration to practical purposes), until you have learned thoroughly the properties of triangles. We shall not, however, attempt to treat of this subject here, but shall simply explain a few names which are commonly used in reference to different triangles, and then point out to you a very important truth in reference to every right-angled triangle.

B

Triangles are sometimes classified according to the comparative lengths of their sides, and are then said to be either equilateral, i.e., having all three sides equal (as A), or isosceles, i.e., having two sides equal (as B), or scalene, i.e., having all three sides unequal (as C).

Triangles are also often classified according to their angles. If a triangle contains one right angle (as D) it is called a right-angled triangle; if it contains one obtuse angle (as E) it is said to be obtuse-angled; but if all three angles are acute (as F) it is an acute-angled triangle.

E

Some of you will have learned already that it is impossible for any triangle to have more than one right angle or one obtuse angle, because the three angles of every triangle, when taken all together, make up exactly two right angles; so that every right-angled and every obtuse-angled triangle must have two acute angles, while every acute-angled triangle has three.

Now, the sides of a right-angled triangle are known by special names, the base (AB) and the perpendicular (BC), which contain the right angle, and the hypothenuse (AC), which is opposite to the right angle.

It is an important truth which some of you have learned already in the first book of Euclid, and

which will help you to solve many problems in mensuration, that in every right-angled triangle the square upon the hypothenuse is equal to the sum of the squares upon the base and perpendicular.

From this we get two valuable rules which enable us, when we have given

B

any two sides of a right-angled triangle, to find the other side:

1. To find the hypothenuse, add the square of the base to that of the perpendicular, and take the square root. 2. To find either the base or the perpendicular, subtract the square of the side given from that of the hypothenuse, and take the square root.

Example 1.-Find the perpendicular of a right-angled triangle whose base is 56 feet and hypothenuse 65 feet.

B

Example 2.-Find the hypothenuse of a right-angled triangle whose base is 85 feet and perpendicular 132 feet.

1. In an isosceles triangle the angle at the vertex is 25° 15′ 16′′, what is each angle at the base?

2. A certain right-angled triangle has one of its acute angles double of the other; find the angles.

3. In an isosceles triangle the angle at the vertex is double each angle at the base; find the angles.

4. Find hypothenuse when base = 51 yds. and perpendicular 67 yds.

=117 ft.

5 ft. 9 in.

86 ft. 8 in.

56 ft. 8 in. 88 ft.

103 ft. 6 in.

6 ft. 3 in.

12. Find base when perpendicular=46 yds. and hypothenuse 530 yds.

13.

125 ft.

14. A field is in the form of a right angle, the base being 1462 yds., and hypothenuse 2138 yds.; how many yards of fencing will be required to go round the field?

15. The side of a square is 1000 feet; what is the length of its diagonal?

16. What is the height of an equilateral triangle, each of whose sides measures 20 feet.

17. The base of an isosceles triangle measures 50 ft., and each side 65 feet; find the perpendicular.

18. A field is in the form of a right-angled triangle, the base being 1584 yds., and the perpendicular 2350 yds.; find the cost of fencing it all round, at 10d. per yard.

19. In a similar field the hypothenuse is 1010 ft., and the base 434 ft.; find the cost of fencing it, at 94d. per yard.

20. In a similar field, the base is 720 ft., and the hypothenuse 962 ft.; find the cost of fencing, at 10d. per yard.

21. A rope, 76 feet long, will just reach from one side of a street to the top of a house 671⁄2 ft. high, exactly on the opposite side; how wide is the street?

22. A man who is carrying down a street a ladder 50 ft. long, rests it upon the road in such a position that it will exactly reach a window 28 ft. high on one side, or another window 36 ft. high on the other side; find the width of the street.

23. The diagonal of a square is 20 yards; find the length of each side.

24. A perpendicular rock 103 ft. high stands close to a river whose breadth is 302.97 ft; find the length of the cord which will exactly reach from the top of the rock to the opposite bank of the river.

LESSON V. THE SQUARE AND RECTANGLE.

The square and rectangle both belong to that class of figures previously mentioned, and called quadrangles; they are both contained by four straight lines, and all the angles in each are right angles; but in the square all the sides are equal (A), while in the rectangle only the opposite sides are equal (B).

5

7

B

3

To find the area of a square or rectangle, multiply two adjacent sides together.

Thus, the area of A is 5 × 5 = 25; and that of B is 7 × 3 = 21. The correctness of this rule may easily be tested experimentally; the two figures above are divided out into a number of small squares, each containing exactly one square unit, and there are twenty-five of these in the square A, and twenty-one in the rectangle B.

This simple rule lies at the basis of all rules in mensuration, used

for measuring surfaces. Whenever a surface (or area) has to be measured, we always multiply two numbers together, just as in calculating volume or solidity we always multiply three numbers together, and then modify the result by introducing another factor so as to get the correct answer. Thus, in one sense, all areas are calculated in reference to the square, and then corrected; and all volumes are calculated in reference to the cube, and then corrected.

What was learnt in the previous lesson (IV.) will enable us now to solve two other interesting problems in connection with squares and rectangles.

1. To find the area of a rectangle when one side and the diagonal are given

For example, in the figure, let us suppose that instead of

d

a

B

A

having given the length of AB, BC (which, for shortness, we will call respectively c and a, from the letters placed at the opposite angles), we have given only a and d. Now the area

can be found in no other way than by multiplying a and c, so that we must begin by ascertaining the length of the latter. Now by the preceding lesson we know that

2. To find the area of a square when its diagonal is given.

Let us, as before, call the diagonal d. Now the area of the square is AB2, or (which is exactly the same), AB × BC; and whatever we do to find the area will simply amount to finding the length of one side and then squaring it. But the preceding lesson has enabled Aus to find the length of one side, for AB2 + BC2 = d2 AB2 = d2 2 AB2 = d2

B

i.e., AB2

+