2. The hypothenuse of a right-angled triangle is equal to the square root of the sum of the squares of the other two sides; and

. 3. Either of the two shorter sides of a right-angled triangle is equal to the square root of the difference of the squares of the hypothenuse and the other side.

Exercises.

1. If the base of a right-angled triangle is 60 feet, and the perpendicular 45 feet, what is the hypothenuse?

OPERATION.

602+452 = 3600+2025 = 5625; √5625 75, feet, Ans. 2. If the hypothenuse of a right-angled triangle is 75 feet, and one of the other sides 60 feet, what is the third side?

7526025625

OPERATION.

3600= 2025; √2025 =

45, feet, Ans. 3. A fort which is 15 feet high is surrounded by a moat 20 feet wide; what must be the length of a ladder that will just reach from the outer edge of the moat to the top of the fort?

Ans. 25 feet.

4. Two men travel from the same place, one due east, and the other due north. One travels the first day 60 miles, and the other 80 miles. How far apart are they at the end of the day? Ans. 100 miles.

5. A line 36 meters long will exactly reach from the top of a perpendicular tower standing on the brink of a river, known to be 24 meters broad, to the opposite bank; what is the hight of the tower? Ans. 26.83+ meters.

6. A tree broken off 30 feet from the ground and resting on the stump, touches the ground 40 feet from the stump; what was the hight of the tree? Ans. 80 feet.

7. The rafters of a house, each 25 feet long, meet at the edge of the roof 15 feet above the attic floor; required the width of the house. Ans. 40 feet.

What is the second Principle? The third?

QUADRILATERALS.

412. A Quadrilateral is a polygon having four sides, and

therefore four angles.

413. A Parallelogram is a quadri- D lateral having its opposite sides parallel.

A RECTANGLE is a right-angled parallelogram; as the figure A B C D.

A

C

B

A SQUARE is a rectangle having equal sides; as the figure EFGH. L

A RHOMBOID is a

F parallelogram

having

no right angles; as the

figure I J K L.

14

K

A RHOMBUS is a rhomboid having

equal sides; as the fig

ure M N O P.

U

T

M

N

414. A Trapezoid

is a quadrilateral having only two of its sides parallel; as the figure R STU.

S

415. A Trapezium is a quadrilateral having no two of its sides parallel; as the figure V W X Y.

AREAS OF TRIANGLES AND QUADRILAT

ERALS.

416. By Geometry, may be proved, in relation to areas, the following

PRINCIPLES.

1. The area of a PARALLELOGRAM is equal to the product of the base by the altitude.

What is a Quadrilateral? A Parallelogram? A Rectangle? A Rhomboid? A Rhombus? A Trapezoid? A Trapezium? To what is the area of a parallelogram equal?

This has been shown to be the case with a rectangle (Art. 210), and that it applies equally to a rhomboid or rhombus, appears from the diagram, in which the rhomboid A B C D is equivalent to the rectangle AB E F, of the same base and altitude.

2. The area of a TRAPEZOID is equal to the product of half the sum of the parallel sides by the altitude.

H

Α

D

For, any trapezoid A B C D is equivalent CK to a parallelogram AL KD of the same altitude, and whose base AL is equal to H I, which is half of A B+C D.

L

I

3. The area of a TRIANGLE is equal to the product of half the base by the Baltitude, or of half the altitude by the

base.

For, any triangle A B C is equivalent to one half of the parallelogram B C E A, of the same base and altitude.

4. The area of a TRAPEZIUM, or of any polygon, is equal to the sum of the areas of B the triangles into which it may be resolved.

Thus, the trapezium W X Y Z is equal to the triangle W X Z plus the triangle X Y Z, made by the diagonal X Z.

E

D

5. The area of a REGULAR POLYGON is equal to the product of the perimeter by half the perpendicular drawn from the center to any one of the sides.

For, any regular polygon, A B C DEF, may be resolved into as many equal triangles as it has sides, by drawing from the center, O, the lines O A, O B, O C,

etc.

F

N

To what is the area of a trapezoid equal? Of a triangle? Of a trapezium? Of a regular polygon?

Exercises.

1. What is the area of a board 18.8 feet long and 2.7 feet wide? Ans. 50.76 sq. ft.

2. What is the area of a board 28 feet long and 15 inches broad? Ans. 35 sq. ft. 3. If the base of a gable of a house be 40 feet long and its perpendicular hight 20 feet, how many square feet of boards will be required to cover it? Ans. 400 sq. ft.

4. How many acres in a triangular lot, one side measuring 32 rods, and the shortest distance from this side to the opposite angle being 14 rods? Ans. 1 A. 64 sq. rd. 5. If the parallel sides of a lot be 75 and 33 yards, and its breadth 20 yards, what is the area in square rods?

Ans. 35.7+ sq. rd.

6. How many hectares in a rectangular meadow 640 meters long and 240 meters wide? Ans. 15 hectares and 36 ares.

7. One of the diagonals of a field in the form of a trapezium is 160 rods long, and the perpendiculars from the opposite angles to that diagonal are 70 and 50 rods; what is the area? Ans. 60 acres.

417. When the three sides of a triangle are given, we may,. to find the area,

Take half the sum of the three sides, subtract therefrom each side separately, multiply together the four results, and extract the square root of the product.

8. The sides of a triangle are 13, 84, and 85 rods, respectively; what is its area? Ans. 3 A. 66 sq. rd.

9. The sides of a certain field in the form of a trapezium measure 30, 35, 40, and 25 rods, respectively, and the diagonal which forms a triangle with the first two sides, 45 rods; what is the area ? Ans. 6 A. 61.8 sq. rd. 10. What is the area of a regular hexagon, whose sides

When the three sides of a triangle are given, how may the area be found?

are each 14.6 feet, and the perpendicular from the center to a side 12.64 feet? Ans. 553.63+ sq. ft.

CIRCLES.

418. A Circle is a plane figure bounded by a curved line, all the points of which are equally distant from a point within, I called the center.

The CIRCUMFERENCE is the bound

ing line; as the line A E B D.

The DIAMETER is any straight line drawn through the center and terminating in the circumference; as the line A B.

The RADIUS is any straight line drawn from the center to the circumference; as the lines CA, CB, or CD.

D

A

B

C

E

419. A Square is said to be inscribed in a circle when the vertices of its angles are in the circumference. Thus,

The square ABCD is inscribed in a circle.

420. By Geometry there may be proved the following

PRINCIPLES.

1. The CIRCUMFERENCE of every circle is nearly 3.1416 times its diameter. Hence,

2. The CIRCUMFERENCE is equal to the product of the diameter by 3.1416; and

3. The DIAMETER is equal to the quotient of the circumfer ence divided by 3.1416.

What is a Circle? The Circumference? The Diameter? The Radius ? How many times the diameter is the circumference? To what is the circumference equal? The diameter ?