329. PROBLEM. To construct a triangle equivalent to a given polygon. Given the polygon ABCDE. To construct the triangle PCo equivalent to ABCDE. In a similar manner draw Co. Then PCO is the required triangle. Proof : PCDE = ABCDE since ▲ APC = ▲ ABC (Why?). Further, PCDE has one less vertex than ABCDE. But APCO = PCDE since ▲ ECD = ▲ ECO (Why?). Hence, ▲ PCO is equivalent to the given polygon. 330. PROBLEM. To construct a rectangle on a given base and equivalent to a given parallelogram. Construction. Let b and h be the base and altitude of the given parallelogram, b' the base of the required rectangle, and x the unknown altitude. Then we are to determine x so that b'xbh, that is, b': bh: x. Hence, x is the fourth proportional to b', b, and h. (See § 291.) Construct this fourth proportional, showing the complete solution. This construction is attributed to Pythagoras. It represents a much higher achievement than the discovery of the Pythagorean proposition itself. 1. Show how to modify the last construction in case the given figure is a triangle. Give the construction. 2. Construct a rectangle on a given base equivalent to a given irregular quadrilateral. 3. Construct a rectangle on a given base equivalent to an irregular hexagon. 4. On a side of a regular hexagon as a base construct a rectangle equivalent to the hexagon. 5. Construct a parallelogram on a given base equivalent to a given triangle. Is there more than one solution? 6. Construct a square whose area shall be three times that of a given square; five times. One half the area; one fifth. 7. Construct an isosceles triangle, with a given altitude h, equivalent to a given triangle. 8. Draw a line parallel to the base of a triangle and cutting two of its sides. How will the resulting triangle and trapezoid compare in area, (a) If each of the two sides of the triangle is bisected? (b) If each of the two sides of the triangle is three times the length of the corresponding side of the trapezoid? 9. Construct a triangle whose base and altitude are equal and whose area is equal to that of a given triangle. 10. In a parallelogram ABCD, any point E on the diagonal BD is joined to A and C. Prove that ▲ BEA and BEC are equivalent, and also that A DEA and DEC are equivalent. 11. The sides of a triangle are 6, 8, 9. A line parallel to the longest side divides the triangle into a trapezoid and a triangle of equal areas. Find the ratio in which the line divides the two sides. 12. Draw a line parallel to the base of any triangle, and cutting two of its sides. How do the altitudes of the resulting triangle and trapezoid compare, (a) If they are equal in area? (b) If the area of the triangle is three times that of the trapezoid? 13. Through a point on a side of a triangle draw a line dividing it into two equivalent parts. SOLUTION. Let P be the given point. Draw the median BD. Draw BE || PD and draw PE. C D This is Ex. 4, § 313, and hence that ▲ EOD = ▲ BOP. 14. Through a given point on a triangle draw a line which divides it into two figures whose areas are in the ratio }. 15. Inscribe a circle in a triangle, touching its sides in the points D, E, F. With the vertices as centers, construct circles passing through these points in pairs. Show that each of these latter circles is tangent to the other two. SUMMARY OF CHAPTER IV. B 1. State what is meant by the area of a rectangle. Give the formula. 2. Give formulas for areas of parallelograms and triangles. 3. How is the formula for the area of a trapezoid obtained? 4. What theorems of this chapter can be stated algebraically, as (a + b)2 = a2 + 2 ab + b2. 5. State the theorem on the ratio of the areas of two similar triangles; two similar polygons. Give examples. 6. Tabulate the problems of construction given in this chapter. 7. If two rectangles have the same base, how does the ratio of their areas compare with the ratio of their altitudes? 8. If two triangles have equal altitudes, how does the ratio of their areas compare with the ratio of their bases? 9. State all theorems of this chapter proved by means of the Pythagorean proposition. 10. State some of the more important applications of the theorems in this chapter. Return to this question after studying the succeeding list of problems. PROBLEMS AND APPLICATIONS. 1. Find the area of a square whose diagonal is 6 inches. 2. Find the area of a square whose diagonal is d inches. 3. ABCD is a square placed at the crossing of two strips of equal width, as shown in the figure. The small black square has two vertices on the sides of the horizontal strips and two on the sides of the vertical strip. (a) Find the area of each square when the width of the strips is 4 inches. (b) Compare the area of the black square and the white border surrounding it. (c) Can squares be placed as in the figure in case the strips are of unequal width? In the two following questions let the small square be drawn with two vertices on the sides of the horizontal strip and one diagonal parallel to these sides. (d) If the horizontal strip is 4 inches wide, what must be the width of the vertical strip in order that the large square may have twice the area of the small one? HINT. The diagonal of the small square is 4 inches. (e) If the horizontal strip is a inches wide, what must be the width of the vertical strip in order that the area of the black square shall be 1 the area of the larger square? n 4. Prove that the area of a rhombus is one half the product of its diagonals. 5. Prove that the area of an isosceles right triangle is equal to the square on the altitude let fall upon the hypotenuse. 6. If the diagonals of a quadrilateral intersect at right angles, prove that the sum of the squares on one pair of opposite sides is equal to the sum of the squares on the other two sides. 7. Inscribe a square in a semicircle and in a quadrant of the same circle. Compare their areas. See Ex. 8, page 147. 8. In the triangle ABC, CD is an altitude. E is any point on CD. If DE is one half CD, compare the area of the triangle AEB and the sum of the areas of the triangles AEC and BEC. Also compare these areas if DE is one nth of DC. 4 D B D G = NP, R 9. ABCD is a square, and E, F, G, H are the middle points of its sides. On EF, points N and Q are taken so that PNPQ. Similarly KL = KM, also KL and so on. (a) Prove that AMOL is a rhombus. (b) If AB - 6 inches and if KM = ME, find the sum of the areas of the four rhombuses AMOL, BQON, etc. = (c) If HE = 8 inches, what is the area of the whole square? (d) What part of KE must KM be in order that the sum of the rhombuses shall be the area of the square? (e) If AB rhombuses shall be the area of the square. = a, find KM so that the sum of the 1 n H M A E (f) Prove that L, O, R lie in the same straight line. 10. ABCD is a square and E, F, G, H are the middle points of its sides. SN PK = QL = RM. = = (a) If AB = 6 inches and if PK 1 inch, find the sum of the areas of the triangles EHN, EFK, FGL, and GHM. (b) If AB = 6, find PK so that the sum of the H four triangles EFK, etc., shall be one fifth of the whole square. = (c) If AB: 6 and if the points E, K, Q lie in a straight line, find the sum of the areas of these triangles. (d) If AB = a and if PK = one nth of PO, find the sum of the areas of the triangles. (e) If AB = a, find PK so that the sum of the triangles shall be one mth of the whole square. What will be the length of PK in case the triangles occupy one half of the whole square? 11. The sides of two equilateral triangles are a and b respectively. Find the side of an equilateral triangle whose area is equal to the sum of their areas. |