Εικόνες σελίδας
PDF
Ηλεκτρ. έκδοση

1. What is the distance from the top of a perpendicular tower 80 ft. in height, to a point 60 ft. from the base of the tower, the latter distance being measured on a line perpendicular to the side of the tower?

ANALYSIS. The distance required is the hypotenuse of a right triangle, in which 60 ft. and 80 ft. are the lengths of the remaining sides. √602+802= √/10000 = 100 ft., hypotenuse.

2. Find the height of the tower in the last example, having given the hypotenuse 100 ft., and the base 60 ft.

ANALYSIS. As the perpendicular equals the square root of the difference of the squares of the hypotenuse and base,

√/1002 — 602 = = 1/6400= 80 ft., height.

3. A mason desiring to know whether the walls of his building were at right angles with each other, measured 12 yards on one wall, and 16 yards on the one which adjoined it, commencing both measurements at the same point and extending them horizontally; what should be the distance apart of the extreme ends of these lines, if the walls are at right angles?

4. Find the hypotenuse of a right triangle, the base being 48 yd. and the perpendicular 36 yd.

5. A flag-pole on the edge of a creek is 150 ft. high, and the distance from the top of the pole to the opposite edge is 250 ft.; what is the width of the creek?

6. What is the area of a right triangle, if the hypotenuse measures 120 ft. and the base 96 ft.?

7. What is the area of a right triangle, if the square of the hypotenuse is 288 chains, and the base and perpendicular are equal in length?

8. What is the area of a right triangle, if the base is 20 chains, and the hypotenuse 1 mile?

QUADRILATERALS.

A Quadrilateral is a plane figure bounded by four straight lines.

Quadrilaterals are divided into classes, as follows:

A Trapezium is a quadrilateral which has no two of its sides parallel.

Note.-Two lines are Parallel, when they are situated in the same plane and have the same direction.

A Trapezoid is a quadrilateral which has only two sides parallel.

A Parallelogram is a quadrilateral which has two pair of parallel sides.

[blocks in formation]

A Rhomboid is a parallelogram whose adjacent sides are not equal, and whose angles are not right angles.

A Rhombus is a parallelogram whose sides are all equal, and whose angles are not right angles.

A Rectangle is a parallelogram whose angles are all right angles.

A Square is a parallelogram whose sides are all equal, and its angles all right angles.

[blocks in formation]

The Altitude of a parallelogram is the perpendicular distance between two of its parallel sides; as the line DE in the figure of the rhombus.

The Diagonal of a figure is a straight line joining its opposite corners; as the line DB in the figure of the

square.

The Perimeter of a figure is the sum of all its sides.

To find the area of a parallelogram.

RULE.

Multiply the base by the altitude.

1. What is the area of a square field, the sides measuring 50 chains each?

50 × 50 = 2500 square chains, area.

2. Find the area of a field in the form of a parallelogram, whose base is 40 rods, and altitude 25 rods.

3. How many acres in a piece of land in the form of a rhombus, the base being 30 chains, and the altitude 22 rods?

4. The base of a rhombus is 30 rods, and the altitude 150 yards; what is the area?

5. The base of a rhomboid is 375 ft. 6 in., and the perpendicular height 40 yards; what is its area?

To find the area of a trapezoid.

RULE.

Multiply half the sum of the parallel sides by the altitude. 1. What is the number of square feet in a trapezoid, one of the parallel sides being 30 inches, the other 4 feet, and the altitude 5 yards?

4 feet 48 inches; 4830-39 inches.

=

2

5 yards = 180 inches; 39x180-48 square feet, area.

2. How many square feet of surface in a board, the ends being parallel and measuring 18 inches and 9 inches respectively, the length being 12 feet?

3. What is the area of a trapezoid whose parallel sides are 225 and 330 yards, and altitude 125 feet?

4. What is the area of a trapezoid whose parallel sides are 20 yards and 16 feet, and altitude 15 inches?

To find the area of a trapezium.

RULE.

Divide the trapezium into two triangles by a diagonal line; the sum of the areas of these triangles will be the area of the trapezium.

1. A meadow in the form of a trapezium, whose sides are 30, 40, 50, and 60 rods, measures 50 rods on a diagonal line dividing the first two sides from the others; what is the area of the field?

ANALYSIS.-Applying the rule for finding the area of a triangle when the three sides are given, we find the area of one of the triangles=160 x 30 x 20 x 10=1/360000=600 square rods; the other=1/80 x 30 x 30 x 20=1/1440000=1200 square rods; and the area of the trapezium=1200+ 600 = 1800 square rods.

2. Given the diagonal of a quadrilateral field, 40 rods, and the altitudes of the triangles into which the diagonal divides the field, 120 feet and 75 feet respectively; what is the area of the field?

3. The sides of an irregular-shaped field are 15, 20, 25, and 30 rods, and the diagonal 100 yards; what is the area?

4. What is the area of a trapezium whose diagonal is 32 feet, the altitudes of the triangles into which the trapezium is divided being 12 and 16 feet respectively?

CIRCLES.

A Circle is a plane figure bounded by a curve, all the points of which are equally distant from a point within it, called the centre.

The Circumference is the curve which bounds the circle.

The Radius is any straight line drawn from the centre to the circumference.

The Diameter is any straight line drawn through the centre and terminated each way by the circumference.

In the figure, O is the centre, the curve ABCD is the circumference, the circle is the space inside of the A curved line, OA and O B are radii,

A B is a diameter.

D

B

To find the circumference of a circle, the diameter

being given.

RULE.

Multiply the diameter by 3.1416.

1. The diameter of a wheel is 6 feet; what is the circumference?

6 x 3.1416 18.8496 feet, circumference.

=

2. If the diameter of the earth is 7960 miles, what is its circumference?

3. Find the circumference of a circular garden whose radius is 500 feet.

4. What is the circumference of a tree whose diameter is 30 feet?

« ΠροηγούμενηΣυνέχεια »