To find the third side of a right triangle.

It is often convenient to know how to find the unknown side of a right triangle when two sides are given. Until the pupil has made a study of geometry, he cannot prove the truth of it except by illustration.

25 FT.

12 FT.

There are two rules needed here; one where one of the short sides is to be found; the other, where the long side, or hypotenuse is to be found.

Remember:

1. To find one of the short sides, (a) Find the square of each known

side.

(b) Find the square root of the difference of these squares.

2. To find the long side or hypotenuse,

(a) Find the square of each known side.

(b) Find the square root of the sum of these squares.

PROBLEMS.

1. Find the third side of the right triangle in the last illustration.

WORK:

(a) 252 = 625. 122 = 144.

-

=

(b) 625 144 481. 481 = 22, almost.

Hence, the third side is almost 22 ft. long.

2. Find the hypotenuse of a triangle whose other two sides are 4 in. and 6 in., respectively.

16+

=

(b) 16 +36 52. √52 = 7.2.

Hence, the hypotenuse is 7.2 inches.

3. I have one fourth of a quarter section of land. I wish to put a fence diagonally across it. How long will the fence be?

NOTE. Illustrate this by diagram.

WORK AND EXPLANATION:

Each side of the land is 80 rds. long. Hence, I know two sides of the right triangle are each 80 rods Then, to find the long side,

Hence, the length of the fence will be a little more than 113.13 rods.

To illustrate that the square on the hypotenuse equals the sum of the squares on the other two sides,

Take a large sheet of paper and on it draw a right triangle whose two shorter sides are 3 inches and 4 inches, respectively. Now from the rule, you find that the hypotenuse is just 5 inches. Next, draw the squares on the sides of the triangle. Divide each square into smaller squares each one inch on a side. Now, count the small squares in each of the three large squares. You find that,

The large square on the shortest side of the triangle has 9 small squares.

The large square on the next short side of the triangle has 16 small squares.

Both the large squares considered have 25 small

squares.

The large square on the hypotenuse has 25 small squares.

NOTE.- What has just been illustrated is known as the Pythagorean proposition, because Pythagoras was the first to demonstrate it. It is often stated,

The square on the hypotenuse equals the sum of the squares on the other two sides.

REGULAR POLYGONS.

A polygon is a plane figure having many sides and many angles.

A regular polygon is one having equal sides and equal angles.

The perimeter of a polygon is the sum of the sides.

Regular polygons may be named as follows:

NOTE.- Let the pupil draw figures to illustrate each of the For this work his compass again becomes valuable.

above.

A regular polygon

can be divided into as many isosceles triangles as it has sides, as shown here.

If a line be drawn from the center. O, to G the middle of the base of any triangle, as to the middle of AB, it represents the altitude of the triangle, or the apothem of the polygon.

Then, to find the area of the polygon, we need only find the area of one triangle and multiply by the number of triangles.

This is equivalent to taking the product of the sum of the bases, or the perimeter of the polygon, by the apothem and dividing by two. Hence, to find the area of any regular polygon,

Divide the product of the perimeter multiplied by the apothem

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CIRCLES.

A circle is a plane surface bounded by a curved line, every point of which is equally distant from a point within, called the center.

bounding line.

The circumference of a circle is its

NOTE. Be careful to avoid giving the pupil the common impression that the circumference is the circle.

A chord is any line which terminates in the circumference without passing through the center.

CIRCUMFERENCE OF A CIRCLE.

The circumference of any circle may be found if the diameter is known, by multiplying the diameter by 3.1416.

Conversely, if you know the circumference of any

circle, you can find the diameter, by dividing the circumference by 3.1416.

NOTE. For ordinary purposes 3 may be used in place of 3.1416. The form is convenient.

take that portion of the circle AOP and lay the circumference, AP on a flat surface, it will look like the first picture here. The smaller the sections the nearer AP will be to a straight line when laid on the flat surface. If half the circle is taken for that and the other half is cut in a similar way, the two parts may be fitted together to form a rectangle like that in the second picture here. The width of

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of the circle, and the length is equal to half the cir cumference.

Then, to find the area of a circle,

Multiply the radius by one half the circumference.

A. H.-29