CHAPTER XXII EQUALITY OF AREAS The two basic methods are: 1. Prove their area formulas equal. 2. Prove the figures congruent. 312. Two parallelograms are equal if they have equal bases and equal altitudes. 313. Two triangles are equal if they have equal bases and equal altitudes. 314. Two trapezoids are equal if they have equal mid-lines and equal altitudes. 315. A triangle equals one-half a parallelogram if they have equal bases and equal altitudes. *316. The square on the hypotenuse of a right triangle equals the sum of the squares on the other two sides. 317. The square on the hypotenuse of a right triangle minus the square on a second side equals the square on the third side. *318. PROBLEM: Construct a square equal to the sum of two given squares. 319. PROBLEM: Construct a square equal to the difference of two given squares. *320. PROBLEM: Construct a triangle equal to a given polygon. 321. How can a2 be represented in Geometry? How can ab? How can 1⁄2 ab? *322. PROBLEM: Construct a square equal to a given parallelogram. SUMMARY 323. State three methods of proving areas equal. 324. EXERCISES 1. The non-parallel sides of a trapezoid form with the parts of the diagonals two equal triangles. 2. The median of a triangle divides the triangle into two equal triangles. 3. The line joining the middle points of two sides of a triangle cuts off a triangle equal to one-fourth of the triangle. 4. The line joining the middle points of two adjacent sides of a parallelogram cuts off a triangle equal to one-eighth of the parallelogram. 5. Divide a triangle into three equal triangles by drawing lines from any one vertex. 6. Construct a square equal to a given triangle. 7. Construct an isosceles triangle equal to a given triangle and having the same base. 8. What is the locus of the vertices of all equal triangles on the same base? 9. The bases of a trapezoid are 16 ft. and 20 ft., and the altitude is 12 ft. Find the base of an equal rectangle having the same altitude. 10. The base of a triangle is 15 ft. and the altitude 8 ft. Find the perimeter of an equal rhombus with altitude 6 ft. 11. Upon the diagonal of a rectangle 12 ft. by 5 ft. a triangle is constructed equal to the rectangle. What is its altitude? 12. Find the side of a square equal to a trapezoid with bases 28 ft. and 22 ft., and each non-parallel side 5 ft. 13. A trapezoid equals a triangle having the same altitude and a base equal to the sum of its bases. 14. A square equals one-half the square on its diagonal. 15. The diagonals of a parallelogram make four equal triangles. 16. An isosceles right triangle equals one-fourth the square on its hypoten use. 17. A right triangle with one angle 60° equals one-half the equilateral triangle on its hypotenuse. 18. Any line through the intersection of the diagonals of a parallelogram divides the parallelogram into two equal figures. 19. If two triangles have equal bases and equal altitudes, lines parallel to the bases and equally distant from the vertices cut off equal triangles. CHAPTER XXIII PROPORTION IN AREAS The basic method is: Divide their area formulas. 325. Two parallelograms are to each other as the products of their bases and altitudes. 326. Two parallelograms with equal bases are to each other as their altitudes. 327. Two parallelograms with equal altitudes are to each other as their bases. 328. Two triangles are to each other as the products of their bases and altitudes. 329. Two triangles with equal bases are to each other as their altitudes. 330. Two triangles with equal altitudes are to each other as their bases. *331. Two triangles, having an angle of the one equal to an angle of the other, are to each other as the products of the sides including the equal angles. *332. Two similar triangles are to each other as the squares of any two corresponding sides. *333. Two similar polygons are to each other as the squares of any two corresponding sides. *334. Two similar regular polygons are to each other as the squares of their radii, or as the squares of their apothems. *335. The areas of two circles are to each other as the squares of their radii. SUMMARY 336. State three methods of proving areas in proportion. 337. EXERCISES 1. Draw a line from a vertex of a triangle to a point in the opposite side which shall divide the triangle into two triangles in the ratio of 3 to 5. 2. The areas of two similar triangles are to each other as the squares of the corresponding altitudes; medians; angle-bisectors. 3. The base of a triangle is 16 ft. and the altitude is 25 ft. What is the area of a triangle cut off by a line parallel to the base and 15 ft. from the base? 4. In two similar polygons, two corresponding sides are 3 and 5. The area of the first polygon is 90. Find the area of the second polygon. 5. If one of two similar polygons is double the other, what is the ratio of their corresponding sides? 6. From a given rectangle, cut off a rectangle whose area is twothirds that of the given rectangle. 7. Two rectangles have equal altitudes, and bases 15 and 3 respectively. What is the ratio of their areas? 8. The equilateral triangle drawn on the hypotenuse of a right triangle equals the sum of the equilateral triangles drawn on the other two sides. 9. Find the side of an equilateral triangle whose area is four times the area of an equilateral triangle whose side is 3 in. 10. The diameters of two circles are 4 ft. and 9 ft. respectively. What is the ratio of their areas? 11. The diameter of a circle is 15 in. Find the diameter of a circle twice as large. 12. If the area of one circle is four times that of another, and the radius of the first is 6 in., what is the radius of the second? 13. The area of a circle is four times the area of the circle drawn on its radius as a diameter. 14. The area of an inscribed equilateral triangle is one-fourth the area of the circumscribed equilateral triangle. 15. The area of an inscribed square is one-half the area of the circumscribed square. 16. The area of an inscribed regular hexagon is three-fourths the area of the circumscribed regular hexagon. 17. The area of an inscribed regular hexagon is equal to twice the area of an inscribed equilateral triangle. REVIEW QUESTIONS 1. What is the unit of measure for areas? 2. State the basic theorem for areas. 3. The proof of the area of a circle depends upon what two theorems? 4. State the formula for the area of a parallelogram. 5. The area of a triangle equals what? 6. The area of a trapezoid equals what? Also in terms of the mid-line and altitude? 7. The area of a regular polygon equals what? 8. The area of a circle equals what in terms of the circumference and radius? Also in terms of π. 9. Art. 316 is what theorem proved once before? 10. The former proof was by algebra. This proof is by what? 11. What geometrical figure does a2 represent? ab? 1⁄2ab? 12. State a method of proving triangles equal but not necessarily congruent. 13. The perimeters of two similar polygons are to each other as what? 14. The areas of two similar polygons are to each other as what? 15. Define Similar Sectors. 16. The ratio of a sector to its circle is determined by what? 17. The circumferences of two circles are to each other as what? 18. The areas of two circles are to each other as what? 19. In what two kinds of right triangle can the sides be found if only one side is known? 20. Represent algebraically the altitude of an equilateral triangle whose side is a; its area. 21. Represent algebraically the diagonal of a square whose side is a. 22. How can a square be constructed equal to a given triangle? 23. How can a square be constructed equal to a given polygon? 24. What is a method of finding the area of an irregular polygon? 25. In proving areas equal, what methods can be used? 26. If the area of a circle equals R2, what does R equal? 27. If the radius of one circle is twice the radius of another circle, what can be said of their circumferences? Their areas? 28. If the number of sides of a regular circumscribed polygon is indefinitely increased, what lines become variables, and approach what lines as limits? 29. How many times has the theorem of limits been used, and in what theorems? 30. In proving areas in proportion, what methods can be used? |