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TYPES OF COLLEGE EXAMINATION QUESTIONS
1. The area of the inscribed equilateral triangle equals one-half the area of the inscribed regular hexagon.
2. The area of a square inscribed in a sector, whose central angle is a right angle, equals one-half the square on the radius.
3. In a circle whose radius is 20, are inscribed an equilateral triangle, a square, and a regular hexagon. Find the perimeter, apothem, and area of each.
4. About a circle whose radius is 10 are circumscribed an equilateral triangle, a square, and a regular hexagon. Find the perimeter and area of each.
5. The circumference of a circle equals 30. Find the circumference of a circle having twice the area of the given circle.
The radius of a circle is 18. What is the radius of a second circle whose circumference is twice as long? What is the radius of a third circle whose area is twice as great?
7. Two concentric circles have their circumferences equal to 15 ft. and 20 ft. Find the area bounded by the two circumferences.
8. Find the area of a sector whose radius equals 15 and whose central angle is 40°.
9. The angle of a sector is 72° and its arc is 22 ft. 10. A square is inscribed in a circle of radius 5. segment cut off by a side of the square.
11. In a circle whose radius is 10, what is the area of a segment whose central angle is 120°? 60°?
12. From a point without a circle of radius 35 two tangents are drawn forming an angle of 60°. Find the area of the figure formed by the tangents and the nearer arc.
13. If the hypotenuse of a right triangle is the base, and three semicircles are drawn, one on each side as a diameter and above the base, the sum of the two crescents thus formed equals the area of the triangle.
14. Prove this in the right triangle 6, 8, 10.
15. If the radius of a circle is 6 in., find the side and area of the inscribed square; of the inscribed regular hexagon.
16. If the diameter of a circle is 14 ft., and the diameter is divided into parts 6 ft. and 8 ft., and on these parts as diameters semi-circumferences are drawn on opposite sides of the diameter, the compound curve thus formed divides the circle in the ratio of 3:4.
What is its area? Find the area of a
17. Find the area bounded by three arcs of 60° and radius 5, if the concave sides of the arcs are turned within.
18. Two circles C and C' are tangent to each other externally and each is tangent to a straight line L. The line of centers makes an angle of 30° with L. If the radius of the larger circle is 3 in., find the radius of the smaller circle.
19. In an isosceles triangle ABC the sides AC and BC are each 25 in. long, and AB is 30 in. long. How far is A from the side BC?
20. The area bounded by two concentric circles equals the area of a circle whose diameter is that chord of the larger circle tangent to the smaller.
21. The radius of a circle is 4 ft. What is the area of that part of the circle outside the inscribed regular hexagon?
22. A square ABCD has side of 24 in. Using A, B, C, and D as centers and radii equal to 8 in., four quadrants are drawn within the Find the perimeter and area of the figure with the corners cut
23. In the same square using A, B, C, and D as centers and radii equal to 12, four arcs are drawn without the square. Find the perimeter and area of the figure bounded by the four arcs.
24. Do the last two exercises but with an equilateral triangle of side 24 in.
25. A square has a side of 4 ft. If semi-circumferences are drawn on each side as a diameter and within the square, find the areas of the four spaces, bounded each by two quadrants.
26. A square has a side of 12 ft. Semi-circumferences are drawn on each side as a diameter and without the square. Find the area of
27. Do the same with an equilateral triangle.
28. Three equal circles are drawn each tangent to the other two. If the common radius is 5, find the area of the space between the circles.
29. Three equal water pipes are so placed that each touches the other two, and a string is tied around them. If the length of the string is 10 ft., find the radius of the pipes.
30. In the figure of ex. 29, if the radius of the pipes is 4 in., find the length of the string around them.
31. A circular arch of masonry, radius 25 ft., rests on two stone piers which are 40 ft. apart. Find the height of the center of the arch above the level of the top of the piers.
32. The sum of the areas of two similar triangles is 255 sq. in., and the ratio of their sides is 1: 4. Find the area of each.
33. A roadway 60 ft. wide is cut through the middle of a circular field 120 ft. in diameter. Find the area of the remainder of the field.
34. A circular grass plot 12 ft. in diameter is cut by a straight path 3 ft. wide, one edge of which passes through the center of the plot. What is the area of the remaining grass plot?
35. A quarter-mile race track has parallel sides and semi-circular ends of radius 105 ft. Find the length of the parallel sides.
36. If the diameter of the earth is 8,000 miles, how far can the light of a lighthouse 150 ft. high be seen?
37. The minute hand of a clock is 2 in. long. How far does the end of it move in 25 minutes?
38. A and B are two points on a railway curve which is an arc of a circle. If the length of the chord AB is 200 ft. and the shortest distance from the midpoint of the curve to the chord is 4 ft., find the radius of the arc.
39. A right triangle ABC has a fixed hypotenuse AB, 2 in. long. The vertex of the right angle is allowed to take all possible positions. Construct accurately the locus of the mid-point P of the leg AC.
40. Through a point A within a circle whose center is B, chords are drawn. Construct the locus of the mid-points of these chords.
41. A chord of length 8 in. moves with its extremities always on the circumference of a circle whose radius is 5 in. The chord is trisected by the points P and Q. What is the locus of P? Find the length of this locus.
42. A circle of radius 2 in. rolls around a square whose side is 4 in. Construct the locus of the center of the circle and find to two decimal places both the length of the locus and the area enclosed by it.
43. Construct the locus of the circle, radius 11⁄2 in., which rolls around an equilateral triangle whose altitude is 2 in. Compute to two decimal places the length of the locus and the area enclosed by it.
44. It is desired to construct a half-mile race track. The start and finish are to be straight ways intersecting at right angles at the goal. The rest of the track is to be an arc of a circle tangent to the two straight ways. Find the radius of the arc and the length of the arc in feet; also the area enclosed by the track.
45. A plot of ground consisting of two parallel straight sides and two semi-circular ends can be inscribed in a rectangle so that the straight sides of the plot lie in the longer sides of the rectangle. If the distance around the plot is 1⁄2 mi., and the longer sides of the rectangle are each 1,000 ft., find the distance between the parallel sides of the plot, and the length of those sides.
Plane Geometry is principally concerned in:
1. Equality of lines, angles, and areas.
2. Congruent triangles, parallelograms, and circles. Parallel lines, derived from equal angles.
4. Perpendicular lines, derived from equal angles.
5. Inequality of lines and angles.
6. Measurement of lines, angles, and surfaces.
7. Proportion of lines, angles, and areas.
8. Similar triangles and polygons.
All of these involve one thing — equality.
Seven theorems have varying proofs that may not fit all text-books. These seven are 35, 36, 37, 38, 61, 122, and 182. They are either proved in these notes or have a reference letter showing where a logical proof can be found.
(The letters after a note mean that a logical proof can be found as follows: W. Wentworth and Smith. Wl. Wells. S. Schultze and Sevenoak.)
The theorems in this chapter may be treated informally.
7. a. See Summaries. This general axiom is one of the three basic axioms of geometry. The other two are Art. 20 and
20. One of the three basic axioms of geometry. 21. Two cases; when the point (P) is on the line and when the point (P) is outside the line. These are true in plane geometry from the nature of an angle.
22. This is sometimes worded, "Two straight lines cannot enclose a space." If this were true, there would be two straight lines between two points, which is impossible, by 20.
23. From the definition of a straight line, they will coincide if superposed.
24. From 23 and 7ƒ and 21. Two cases.
25. This is Euclid's famous parallel postulate. It is one of the three basic axioms of geometry.
26. From the definition of supplement.
27. From the definition of supplement and of a straight angle.
The basic theorem for equal angles is 23, all straight angles are equal. Informal proofs of two or three statements can be used.
30. This is derived from 23 and 7ƒ.
31. From 30 and 7d.
32. From 23 and 7d.
Similar to 32.
35. The lines cannot meet by 21. A formal proof could be used here, but this can wait till Chapter VI is begun.
36. If the angles were equal, the lines would be parallel by 35. This must be the same as the given figure by 25. A formal proof could be used here but this can wait until Chapter VII is begun.
From 36 and 33.
38. From 33 and 35.
All theorems of angle-sums depend either directly or indirectly upon Art. 26 as a basis. It is sometimes more convenient, however, to use 42.
41. Method is 26. That is, use 26.
42. Method is 26; extend one side, or draw a line through a vertex parallel to the opposite side.
This theorem is useful in proving many other theorems.
43. Method is 26; same as 42.
44. Method is 42 and 7d.
45. Method is 42; draw a diagonal.
47. (n-2) triangles.
48. Method is 42 and 7c.
49. Method is 26, 48, and 7d.
Congruent figures will coincide if superposed, hence superposition should be attempted at first. This is necessary in the first two theorems, Arts. 52 and 53. These may be called the basic theorems of congruent triangles. In the remaining theorems of this chapter,