MENSURATION FOR BEGINNERS

Lesson No. 1. Right-Angled Triangles

Mensuration tells us how to find the lengths of lines, the areas of surfaces, and the volumes of solid figures. Geometry and trigonometry establish certain rules and formulæ. Mensuration teaches their use and application.

When one straight line crosses another so as to form four equal angles, each of these angles is called a right angle. An angle smaller than a right angle is called an acute angle, and an angle larger than a right angle

is called an obtuse angle.

A triangle is a plain figure formed by three straight lines. A right-angled triangle is a triangle which has a right angle. The three angles of a triangle if placed together will make two right angles.

The object of this lesson is to show how to find the length of a side of a right-angled triangle when the lengths of the other two sides are known.

C = hypotenuse.

A = base.

B= perpendicular.

It is proved in geometry that the square of the hypotenuse is equal to the sum of the squares of the two sides.

(See Proposition 47 in "A First Course in Geometry.") That is,

Now suppose that A3 in. and B = 4 in., then we have

A=√52-42√25-16=√9=3.

B=√52-33=√25 - 9=√16=4.

The square of a number is the product of the number multiplied by itself. The square root of a number is that smaller number which multiplied by itself will give that number. Thus, 25 is the square of 5; 4 is the square root of 16.

A subsequent lesson will show you how to find the square root of large numbers. In the exercises which follow, the square root will readily be seen. (See also Lesson 28 in "Mechanics' Bids and Estimates.")

EXERCISES

1. The sides forming the right angle of a right-angled triangle are 6 in. and 8 in. Find the length of the other side.

NOTE. - Square 6 and square 8; add together; take the square root of the sum.

2. The sides forming the right angle are 15 ft. and 8 ft. Find the other side.

3. The sides forming the right angle are 9 ft. and 12 ft. Find the other side.

4. The hypotenuse is 60 in. and the base 48 in. Find the perpendicular.

5. A ladder whose foot is placed on the ground 9 ft. from the front of a house reaches a window at a height of 12 ft. What is the length of the ladder?

6. A ladder 29 ft. long is placed so as to reach a point in the front of a house 21 ft. above the ground. How far is its foot from the house?

7. ABC is a triangle, and from A a perpendicular AD is drawn to the base BC. AD 12 in., BC= 25 in., and BD9 in. Find the lengths of AB and AC.

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Lesson No. 2. Rectangles and Squares

A rectangle is a four-sided figure containing four right angles; that is, it is a figure whose opposite sides are equal and parallel and whose angles are right angles. When the four sides of a rectangle are of equal length the figure is called a square.

The figure ABCD is a rectangle. The line CB is called its diagonal. The diagonal divides the rectangle into two right-angled triangles.

To find the area of a rectangle we multiply the length by the breadth. If ABCD is 8 in. long by 5 in. wide, it will

contain 8 × 5, or 40 sq. in. The proof of this is evident from the figure. If the figure is a square, we find the area by squaring one of the sides; that is, by multiplying it by itself.

If we know the area of a rectangle and the length of one of the sides, we can find the length of the other side by dividing the area by the length already known. If we know the area of a square, we can find the length of a side by taking the square root of the area.

EXERCISES

1. Find the area in square yards of a rectangle whose length is 93 ft. and whose breadth is 27 ft.

2. A rectangle is 36 in. long; it is one-fourth as wide as it is long. Find its area in inches.

3. The perimeter (the distance around) of a square is 28 ft. Find its area in square feet.

4. The diagonal of a rectangle is 15 ft. and the shorter side is 9 ft. Find the area.

5. The diagonal of a rectangle is 29 ft. and one of the sides is 20 ft. Find the area.

6. The area of a square is 169 sq. ft. in feet.

Find its perimeter

7. How long will it take a man to walk around a square containing 40 acres, at the rate of 4 mi. an hour?

NOTE. There are 160 sq. rd. in an acre, and a mile is 320 rd. long.

8. The area of a rectangle is equal to the sum of three squares whose sides are 18 ft., 19 ft., 20 ft. If one side of the rectangle is 31 ft., find the other side.

9. A rectangle is three times as long as it is wide. Its area is 243 sq. ft. Find its length in feet.

10. Find the length of a rectangle whose breadth is 9 ft. and whose area is equal to that of a square the diagonal of which is √288.

In the triangle ABC the perpendicular AD, drawn from A to the opposite side BC, is called the altitude of the triangle relative to BC as base.

To find the area of a triangle, having given one side and the perpendicular drawn to it from the opposite angle or vertex, we multiply the perpendicular, or altitude, by the given side, and take half the product.

You will notice that the triangle ABD is one-half of the rectangle EBDA, and the triangle ADC is one-half of the rectangle ADCF, therefore the sum of these triangles or ABC is one-half of the sum of the rectangles or EBCF.