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CHAPTER VII.

CUMULATIVE REVIEW NO. 2.

1. How many boundaries has a parallelopiped? NOTE. The boundaries of a solid are called Surfaces.

2. What forms the boundaries of a surface?

3. What are the extremities of lines?

4. A point is that which has position without length, breadth, or thickness. Did you ever see a point?

5. A line has length without breadth or thickness. How do we represent lines?

6. A surface has length and breadth without thickHow many surfaces has a cube? How many has

ness.

a sphere?

7. What name is given to those geometric forms that have length, breadth, and thickness?

NOTE.

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A geometric form having length, breadth, and thickness is called a Solid.

QUERY. Is the word "solid" used here in its ordinary sense?

8. If a point were moved in one direction through space, what would its path be?

9. What would be the path of a line which was moved through space in the direction of its length? What would be the path of a line moved in the direction perpendicular to its length?

10. Move a piece of paper through space in such a way that the path of one of its surfaces would be a geometric solid.

11. Cut out a square and lay it on the desk. If it were to rise in the air keeping itself always parallel to the desk, what geometric solid would it form when it had reached a height equal to one of its sides? What would you call it when it had reached a height greater or less than one of its sides?

12. If a circle should rise in the same way, what geometric solid would be formed?

NOTE. Such a geometric solid is called a Cylinder.

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13. Cut out a rectangle whose longer sides are each 22 inches and shorter sides each 8 inches. Place the shorter sides together in such a way that the longer sides form parallel circumferences. If circles were fitted to these circumferences, what kind of a solid would be formed? What would be the area of the curved surface? What is the length of the longest line that can be drawn on either of its plane surfaces?

14. In what geometric form are measures of bushels, half bushels and pecks generally made?

15. If the diameter of the bottom of a cylindrical tin pail is 10 inches, what is its circumference? If its height is 10 inches, how many square inches of tin were used in making its curved surface?

16. A straight line is one that does not change its direction at any point. What is a curved line?

17. Classify lines and define each class.

18. Classify angles and define each class.

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19. What is the complement of an angle of 48° 50'?

20. Work Ex. 19, substituting "supplement" for complement.

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21. How many degrees are there in angle x if its adjacent supplementary angle is 50° larger?

22. How many degrees are there in an angle whose complement is 10° more than 3 times as large?

23. How many degrees are there in each of the three angles formed on the same side of a straight line at a given point of that line if the first is 10° greater than the second, and the second 10° greater than the third ?

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28. If the area of the rectangle = 70 square inches, what is the value of x?

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29. Given, AE 6 inches; AB=7 inches; area of rectangle = 66 square inches; required BC.

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30. Find the area and perimeter of a rectangle each of whose long sides is 7 centimeters more than three times as long as a short side, and the sum of two adjacent sides of which is 27 centimeters.

31. Given a rectangle 12 millimeters long and 10 millimeters wide; find the area and perimeter of a rectangle drawn within it, having its sides parallel to the sides of the original rectangle and at a distance of 2 millimeters from them.

QUERY. - How much shorter is each side of the small rectangle than the homologous side of the large rectangle?

32. A rectangular garden 48 feet by 36 feet has a border 3 feet wide within its limits. How much surface is inside the border? How much is occupied by the border?

33. A square garden, whose dimensions are 8 yards, has a square flower bed in its center whose area is that of the garden, and whose sides are parallel to those of the garden. How long is one side of the flower bed, and how far is the middle point of one of its sides from the middle points of the parallel sides of the garden?

34. What is the sum of two lines 8 feet and 5 feet? Draw a figure on the scale of a centimeter to the foot showing the sum of the squares of those lines, and another showing the square of their sum.

35. Find the area of the rectangles and small square which must be added to a 10-inch square to form a 15inch square; to form an 18-inch square.

36. Given a 10-inch square to make into a larger square. Find by trial how wide the additions must be in order that the rectangles and small square shall contain 96 square inches.

37. Given a 10-inch square as a basis for a larger square. Find by trial how wide the additions must be that they may contain 69 square inches.

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38. Given a 20-inch square to build a larger one. by trial the width of the necessary additions if they contain 176 square inches.

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39. Find the length of x when the sum of the three additions to the 20-inch square is 384 square inches.

40. If x represents the side of a square, what represents the area of the square?

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41. If a2 represents the area of a square, what represents a side?

42. Find the area of a square whose perimeter is 100 inches.

43. Find the perimeter of a square whose area is 64 square inches.

44. Divide a semicircle into two sectors, one of which is 5 times the other. How many degrees are there in the

angle of each?

45. The diameter of a circle is 42 inches. Find the length of the arcs of two sectors which compose the circle, the greater being 10 times the less. Illustrate.

46. An arc which is a quadrant is 16 feet. Find the diameter.

47. What angle is the greatest common measure of a given angle and its complement, if the complement contains 24° more than the given angle?

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48. What angle is the G. C. M. of a given angle and its supplement, if the supplement is 30° more than twice the given angle?

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