164. SIMILAR SURFACES.

1. What is the area of a square, one side of which is 1 inch? 2 inches? 3 inches? 4 inches? 5 inches?

2. If the area of a square is 1 inch, what will be the area of a square the side of which is twice as long? 3 times as long? 4 times as long? 5 times as long?

3. What is the area of a triangle whose base and altitude are 2 feet and 3 feet respectively? 4 and 6 ft.? 6 and 9? 8 and 12? 10 and 15? 20 and 30?

4. If the area of a triangle is 3 feet, how many times as large will be the area of one the sides of which are twice as long? 3 times as long? 4 times? 5 times?

5. What is the area of a rectangle whose length and breadth are 3 and 4 respectively? 6 and 8? 9 and 12? 15 and 20?

6. If the area of a rectangle is 12, how many times as large will be the area of a rectangle whose base and altitude are twice as long? 3 times? 4 times? 5 times?

NOTE 1. Similar rectangles or triangles, or other rectilinear figures, are those whose corresponding dimensions are proportioned; thus, in No. 5 above, the length and breadth of each rectangle are in the ratio of 3 to 4. In No. 3 the bases and altitudes of the triangles are in the ratio of 3 to 4. They are therefore similar. All squares are similar to each other; so are all circles. Why?

NOTE 2. If a line is drawn in a triangle, parallel to one of the sides, and meeting the other two sides, it divides those two sides proportionally; the small triangle cut off is, therefore, similar to the whole undivided triangle.

NOTE 3. The areas of similar surfaces are in proportion to the squares of their like dimensions. Illustrate this truth by the above examples.

7. Draw upon your slate a square, one side of which shall be one inch. Inscribe in this square a circle; that is, draw in it a circle whose circumference shall touch each side of the square. What is the diameter of the circle? What is the area of the square? The area of the circle is .7854 of an inch. (160, Rule 2.)

8. Draw a square whose side is 2 inches, and as before. What is the diameter of the circle? the square? What is the area of the circle?

inscribe in it a circle What is the area of

9. Draw another square whose side is 3 inches, and ask and answer the same questions as before.

Give, if you can, a reason for the rule for finding the areas of circles; viz., "Multiply the square of the diameter by .7854.”

10. Draw two circles about the same centre, one with a radius of 2 inches, and another with a radius of 3 inches. What is the ratio of their areas?

From the above examples and illustrations we may derive the following rules for the use of those who need them.

I. To find the area of any surface which is similar to a given surface.

RULE. As the square of either of the sides or dimensions of the surface whose area is known, is to the square of the corresponding side or dimension of the other surface, so is the area of the first surface to that of the second.

II. To find the side, diameter, or circumference of any surface which is similar to a given surface whose dimensions are known.

RULE. As the area of the figure whose side is known, is to the area of the other, so is the square of any dimension of the former to the square of the corresponding dimension of the latter. Extracting the square root of this fourth term will give the answer.

11. If a pipe 1 inch in diameter discharge 2 gallons of water per minute, how many gallons will be discharged by a pipe 2 inches in diameter? 2 inches? 3 inches? 34 inches ?

12. If a 1 inch tube discharge 20 gallons in 18 minutes, how large a tube will be needed to discharge 80 gallons in the same time? 180 gallons? 320 gallons?

13. There is a right-angled triangle 12 feet in perpendicular height. How far from the base must a line be drawn parallel to the base, to cut off of the whole triangle? (Note 2.) 14. How would a line parallel to the base, and 6 feet from it, divide the triangle?

15. If a rope 6 inches in circumference consists of 450 threads or strands, required the number of such threads to make a 14 inch cable.

16. If 30 feet in length of a cable 10 inches in circumference weighs 117 pounds, how much will 30 feet of another cable of the same stock weigh, that is 15 inches in circumference? What must be the circumference of a rope which would weigh as much as the first cable?

17. There is a triangle containing 85 square rods, and one

of its sides measures 15 rods. What is the area of a similar triangle, the corresponding side of which measures 10 rods? 18. If a perpendicular pole 10 feet long casts a shadow 7}} feet long, what is the height of a steeple that casts a shadow of 140 feet at the same time?

19. A triangular board is 18 inches wide at the base, and 12 feet long. At what distance from the base must a line be drawn to cut off half of it? of it? of it?

165. MISCELLANEOUS EXERCISES IN SURFACES.

1. What is the side of a square floor containing 1521 square feet?

NOTE 1. To find the side of a square equal in area to any given superficies,

Extract the square root of the given area.

2. The side of a square is 15 feet. How long is the diagonal? 3. The diagonal of a square is 20 feet.

of the square?

What is one side

4. A rectangle is 25 feet long and 20 feet wide. What is its area? How long is its diagonal?

5. Two ships have sailed from the same port; one north 50 miles, the other east 60 miles. How far apart are they?

6. Four boys attend the same school. Charles lives 150 rods north from the school-house, James 100 rods east, William 75 rods south, and Henry 200 rods west from the schoolhouse. How far, in a direct line, do Charles and James live apart? James and William? William and Henry? Henry and Charles? Charles and William? James and Henry?

7. A carpenter, wishing to test the correctness of his "square," makes a mark on one of its arms 6 feet, and also another 8 feet, from the point where the two arms meet. How far apart are these marks, if the arms are exactly perpendicular? 8. What is the area of a circular plot of land, the diameter of which is 18 rods? 36 rods? 9 rods? 3 rods?

9. What is the diameter of a circular field, the area of which is 16 acres? 64 acres? 4 acres? 1 acre?

10. A rectangular court is 100 feet long and 20 feet wide. How much further does he travel who goes from one corner to the opposite one, following the direction of the fence, than he who crosses it diagonally?

11. A rectangular field is 15 rods long and 12 wide. What is it worth, at $180 per acre? How much will it cost to fence it, at $2.50 per rod? How far from either corner is the centre of the field?

12. If a leaden pipe, 1 inch in diameter, will fill a cistern in two hours, what will be the diameter of another pipe, to fill the same cistern in one hour?

13. If 20 feet of iron railing weigh 1120 lb. when the bars are 1 inch square, what will 30 feet weigh, if the bars are inch square? What will 50 feet weigh, and what will it come to at 64 cents per pound?

14. If a round pillar 7 inches in diameter has four cubic feet of stone in it, what must be the diameter of a pillar of equal length, to contain ten times as much stone?

15. A gentleman has a fish-pond in the form of a triangle, containing 480 poles; he wants another, one half as large, in the form of a square. Required the side. (Note 1.)

16. There is a rectangular field 220 yards long and 22 yards broad. What length of fence will enclose the same area in a square?

rd.;

17. A farm consists of 4 fields; the first, 2 A. 3 R. 14 sq. rd.; the second, 3 A. 1 R.; the third, 1 A. 0 R. 18 sq. the fourth, 4 A. 3 R. 24 sq. rd. What shall be the side of a square field equal in area to all four?

18. The length of a line, stretched from the top of a steeple to a station 250 feet from its bottom, was found to measure 330 feet. What was the height of the steeple?

19. The breadth of a building is 32 feet, and the height of the angle for the roof, that is, of the ridge above the beams, is 9 feet. Required the length of the rafter.

20. What is the perpendicular of an equilateral triangle, each side of which is 144 yards?

21. A carriage wheel is to be 5 feet in diameter; what will be its circumference? How long must each felloe of the wheel be, if there are to be 6 felloes? How long, if there are to be 5 felloes?

22. If the diameter of a circle is 1 inch, how long will be the diagonal of an inscribed square? What will be the side of the square? What will be the side of an inscribed square, if the diameter is 42 inches?

NOTE. The square root of one half of the square of the diagonal will be one side of the square. Why?

23. The radius of a circle is 34 inches. What is the area of

an inscribed square?

24. Required the difference in area between a circle whose diameter is 24, and a square inscribed in the same circle.

25. What is the difference in area between a square whose equal side is 24, and the largest circle that can be inscribed in the same square?

QUESTIONS. What is a point? a line? a straight line? a curve line? What are parallel lines? oblique or inclined lines? What is an angle? the vertex of an angle? What are right angles? perpendicular lines? horizontal lines? vertical lines? What is an oblique angle? an obtuse angle? an acute angle? a surface? a plane surface a triangle? the height or altitude of a triangle? the base of a triangle? Give examples of each. What is an equilateral triangle? an isosceles triangle? a scalene triangle? a right-angled triangle? the hypothenuse? an obtuse-angled triangle? an acute-angled triangle? a quadrilateral or quadrangle? a diagonal? a parallelogram? a rectangle? a square? a rhomboid? a rhombus? a trapezium? a polygon? What is a polygon of 5 sides called? of 6 sides? of 7 sides? What is a heptagon? an octagon? a regular polygon? an irregular polygon? What is the perimeter of a figure? What is a rectilinear figure? a circle? the diameter of a circle? the radius? a tangent? an arc of a circle? a chord? a segment? a sector? a semicircumference? a semicircle? How is the circumference of every circle supposed to be divided? How many degrees in a semicircle? in a quadrant? What is an ellipse? What is the longer axis called? the shorter?

To what is the area of a square equal? the side of a square? What is the rule for finding the area of a rectangle? of any parallelogram? of a trapezoid? of a triangle? Another rule? How may the area of any surface which is bounded by straight lines be found? What is the rule for finding the area of a regular polygon?

To what is the circumference of a circle equal? How may the area be found? How may the diameter be found, when the area is given? What is the rule for finding the area of an ellipse?

To what is the hypothenuse of a right-angled triangle equal? To what is the hypothenuse equal? To what is either side equal? What are similar rectilinear figures? What figures are always similar? How does a line drawn parallel to one side of a triangle and meeting the other two sides, divide those sides? To what are the areas of similar figures proportional? Give a reason for the rule for finding the area of circles, viz.: "Multiply the square of the diameter by .7854." What is the rule for finding the area of a surface that is similar to a given surface? What is the rule for finding the dimensions of a surface that is similar to a given surface whose dimensions are known?