Miscellaneous Exercises Ex. 103. A road 60 ft. wide runs from one corner to the opposite corner of a square field measuring 500 ft. on a side, the diagonal of the field running along the center of the road. What is the area of that portion of the field occupied by the road? (Carry out the results to two decimal places.) Ex. 104. What is the length of the side of an equilateral triangle equal to a square whose side is 15 in.? Ex. 105. From one vertex of a parallelogram, draw lines dividing the parallelogram into three equal parts. Ex. 106. The sides AB, BC, CD, and DA of quadrilateral ABCD are 10, 17, 13, and 20 respectively, and the diagonal AC is 21. Find the area of the quadrilateral. Ex. 107. If diagonals AC and BD of trapezoid ABCD intersect at E, then ▲ AEB =▲ DEC. (BC and AD are the bases of ABCD.) Suggestion. - Compare ▲ ABD and ▲ ACD. Ex. 108. If X is any point in diagonal AC of □ ABCD, then ΔΑΒΧ=ΔΑΧΟ. Suggestion. Draw the altitudes from B and D to base AX. Ex. 109. If E and F are the mid-points of sides AB and AC of ▲ ABC, then ▲ AEF = } ▲ ABC. Ex. 110. If E is any point within ABCD, then ▲ ABE + ▲ CDE equals the parallelogram. Suggestion.-Through E draw a line parallel to AB. Ex. 111. If ▲ A of ▲ ABC is 30°, prove that the area of ▲ ABC = AB × AC. Suggestion. Draw BD 1 AC. Recall Ex. 128, Book I. Ex. 112. Prove that the area of a rhombus is one half the product of its diagonals. Ex. 113. If E is the mid-point of CD, one of the non-parallel sides of trapezoid ABCD, prove that ABE ABCD. Suggestion. Through E, draw a line parallel to AB. Ex. 114. Construct an isosceles triangle equal to a given triangle, having given one side of length m. Suggestion. Use m as the base. Determine the altitude to m as in Ex. 78, Book IV. Then follow § 241. Ex. 115. Draw through a given point in one base of a trapezoid a straight line which will divide the trapezoid into two equal parts. Ex. 116. If the diagonals of a quadrilateral are perpendicular, the sum of the squares on one pair of opposite sides of the quadrilateral equals, the sum of the squares on the other pair. Note. Supplementary Exercises 42 to 46, p. 298, can be studied now. Review Questions 1. Define area of a plane figure. 2. Distinguish between congruent, similar, and equal figures. 3. State the rule for determining the area of: (a) a rectangle ; (b) a parallelogram ; (c) a triangle; (d) a trapezoid. 4. State the formula for the area of any triangle in terms of its sides a, b, and c, and the number s. What is the number s? 5. State the formula for the area of an equilateral triangle in terms of its side s. 6. State the corollaries by which the areas of two rectangles are compared : (a) If the rectangles have equal altitudes. (b) If the rectangles have equal bases. (c) When no known relation exists between the altitudes or the bases. 7. State the corresponding corollaries for two parallelograms. 8. State the corresponding corollaries for two triangles. 9. State a theorem connecting the areas of a triangle and a parallelogram having equal bases and equal altitudes. 10. State a theorem connecting the areas of two similar polygons. BOOK V REGULAR POLYGONS. MEASUREMENT OF THE CIRCLE 355. Review the definitions given in § 125, § 128, and § 178. 356. A Regular Polygon is a polygon which is both equilateral and equiangular. The figures below illustrate some uses of regular polygons: Notice the regular triangles, hexagons, squares, and octagons. Ex. 1. Prove that the exterior angles at the vertices of a regular polygon are equal. Ex. 2. What is the perimeter of a regular pentagon one of whose sides is 7 in. ? of a regular octagon one of whose sides is 6 in. ? Ex. 3. In § 154, we have proved that the sum of the angles of any polygon having n sides is (n-2) st. 4. How large is each angle of a regular polygon having: (a) 3 sides? (b) 4 sides? (c) 5 sides? (d) 6 sides? (e) 8 sides? (f) 10 sides? Ex. 4. (a) Four square tile can be used to cover the space around a point. (Why?) (b) In the shape of what other regular polygon can tile be made in order that the surface around a point can be completely covered by using tile of the same shape? 357. Each angle of a regular polygon having n sides is n- st. . (See Ex. 3.) n PROPOSITION I. THEOREM 358. A circle can be circumscribed about any regu lar polygon. E Hypothesis. ABCDE is a regular polygon. Conclusion. A circle can be circumscribed about ABCDE.. Proof. 1. A O can be constructed through A, B, and C. Let O be its center and OA, OB, and OC be radii of it. 2. It can now be proved that this circle passes through D by proving OD=0A. (Draw OD.) Suggestions. 1. Compare ABC and ≤ BCD; 21 and 22; then ≤3 and 24. 2. Prove ▲ AOB ≈ ▲ OCD, and then OD = OA. 3. Hence the O passes through D. 3. Similarly the circle can be proved to pass through E. 4. Hence a O can be circumscribed about ABCDE. 359. Cor. 1. A circle can be inscribed in any regular polygon. Proof. 1. AB, BC, CD, etc. are equal chords of the circle which can be circumscribed about ABCDE. 2. Hence these sides are equidistant from 0. Why? 3. Hence a circle can be drawn tangent to each of the sides of ABCDE. D 360. The Center of a regular polygon is the common center of the circumscribed and inscribed circles; as 0. The Radius of a regular polygon is the distance from its center to any vertex; as OA. The Apothem of a regular polygon is the distance from its center to any side; as OF. The Central Angle of a regular polygon is the angle between the radii drawn to the ends of any side; as AOB. The Vertex Angle of a regular polygon is the angle between two sides of the polygon. 360° 361. Cor. 2. The central angle of a regular n-gon is n 362. Notation. The following notation will be employed: (a) 84, 86, or sn will denote one side of a regular inscribed polygon of 4, 6, or n sides respectively. (b) as α or α, will denote the apothem of a regular inscribed polygon of 4, 6, or n sides respectively. (c) P P or Pn will denote the perimeter of a regular inscribed polygon of 4, 6, or n sides respectively. (d) k, ke or kn will denote the area of a regular inscribed polygon of 4, 6, or n sides respectively. To denote the corresponding quantities for a regular circumscribed polygon, a capital letter with the appropriate subscript will be employed. Thus, Sone side of the regular circumscribed pentagon. = A, the apothem of the regular circumscribed pentagon. P, the perimeter of the regular circumscribed pentagon. K, the area of the regular circumscribed pentagon. = = Ex. 5. Find the number of degrees in the central angle and in the vertex angle of a regular polygon of: (a) 3 sides; (b) 4 sides; (c) 5 sides; (d) 6 sides; (e) 8 sides; (f) 10 sides. Find also the sum of the central angle and the vertex angle in each case. Do the results suggest any theorem ? Ex. 6. Prove that any radius of a regular polygon bisects the angle to whose vertex it is drawn. Supplementary Exercises 1 to 2, p. 299, can be studied now. |