We have seen that any kind of an area having straight lines as sides, and right angles as angles, can be divided into rectangles, and thus the whole area found. We are now to deal with triangles and other areas bounded by straight lines, but whose angles are not all right angles. 1. By cutting, compare the two triangles into which a diagonal divides a rectangle. 2. What is the relation of a to b in the rectangle above? Then what part of the rectangle is the triangle? 3. If the dimensions of rectangle ABDC are 10 in. by 6 in., what is the area of triangle ABC? 4. In II compare c with d; e with f. Compare triangle EFG with rectangle EFIH. 5. Compare the base and altitude of triangle EFG with the dimensions of the rectangle. 6. If the dimensions of the rectangle are 12 ft. and 8 ft., what are the base and altitude of the triangle? What is its area? 7. By cutting, show that in an obtuse-angled triangle, as in III, the triangle is also equal to half a rectangle having the same dimensions as the base and altitude of the triangle. It is now seen that Any triangle is equal to half a rectangle having the same dimensions as the base and altitude of the triangle. 8. What is the area of a triangle with base 20 ft. and altitude 12 ft.? UNIV. OF MEASUREMENT OF TRIANGLES, TRAPEZIUMS, ETC. 99 Oral and Written 1. What is the area of a triangle whose base is 12 rods and altitude 14 rods? 2. State your method of finding the area of a triangle. Find areas of triangles of the following dimensions: 3. Base 40 ft., alt.=18 ft. 4. Base 60 ft., alt.=25 ft. 5. Base-3 ft., alt.=9 in. 6. Base 4 rd., alt.=7 ft. 7. 19 ft. and 21 yd. 8. 38 rd. and 224 ft. 9. 64 yd. and 13 ft. 10. 3rd. and 6 yd. 11. Draw or cut out a trapezium, or a quadrilateral no two of whose sides are parallel. 12. Separate it into two triangles A along one of its diagonals, as AB. 13. Find the dimensions of each triangle and its area. 14. What will the area of the trapezium be? B 15. The diagonal of a trapezium is 24 inches; the altitudes perpendicular to it are 18 inches and 15 inches respectively. What is the area? 16. The diagonals of a given trapezium cross at right angles. The point of intersection is 50 feet from the upper end of each diagonal. One diagonal is 100 feet long, the other 150 feet. Find the area. (Draw a diagram.) 17. In right triangles the two sides including the right angle, are called the legs. If one leg is the base, the other is the altitude. Why? 18. The legs of a right triangle are 10 and 15 ft. Find the area. 19. The diagonal of a trapezium is 14 in. The altitudes perpendicular to it measure 6 in. and 8 in. What is its area? 100: MEASUREMENT: OF PARALLELOGRAMS AND TRAPEZOIDS Oral and Written 1. By cutting, find the relation between the two triangles into which any parallelogram is divided by a diagonal. 2. CM, the altitude of the triangle, is also the altitude of the parallelogram, and AB is the base of each. 3. Compare a parallelogram with a triangle having the same altitude and base. D A 4. What kind of parallelogram is shown in the figure? C 5. Compare a rhomboid with a rectangle whose dimensions are the same as the base and altitude of the rhomboid. 6. How is the area of a rectangle found? 7. How, then, may the area of a rhomboid be found when its base and altitude are given? D 16 C 12. To find the area of a trapezoid whose two parallel sides are 24 in. and 16 in. and whose altitude is 12 in. If the trapezoid is divided as in the figure, what are the dimensions of triangle ABC? Its area? If DC is taken as the base of triangle ACD, what is the altitude? The area? What, then, is the area of the trapezoid? 2 A 24. 12 B STATEMENT. of 24 × 12 + of 16 × 12, or of 40 × 12 = 20 × 12, or 240. 13. Notice that since the altitude is the same in each triangle, time may be saved by adding the two bases before multiplying by half the altitude. State the rule, then, for finding the area of a trapezoid. 1. Name 2 quadrilaterals that are not parallelograms. 2. The base and the altitude of a rhombus are each 16 in. Area? 3. The altitude of a rhomboid is of its 2-ft. base. Area is x. 4. One angle of a parallelogram measures 90°. What do the other angles measure? 5. The parallel sides of a trapezoid are 25 and 37 ft., respectively, and the distance between them is 15 ft Required the area. 6. The base of a triangle is 171⁄2 in., its altitude 8 in. What is its area? 7. A lot of land has a frontage of 50 ft. The parallel sides are perpendicular to this frontage. One of its parallel sides measures 10 rd. and the other 80 ft. What is its value at 45¢ a square foot? 8. A triangular park measures 600 ft. on one side, 300 ft. on the other which is perpendicular to it; how many square rods in the park? 9. Compare the area of a rhombus having a base of 16 ft., and an altitude of 12 ft., with that of a 16-ft. square 10. The diagonal of a trapezium is 42 in., and the perpendiculars dropped to it from the angles are 16 in. and 18 in. respectively. Required the area. 13. Base 31 yd., altitude 20 in. 14. Base 2640 ft., altitude mi. is 7 yd. The altitudes perpenWhat is its area? Find the areas of rhomboids: — 11. Base 13 ft., altitude ft. 12. Base 20 rd., altitude 50 ft. 15. The diagonal of a trapezium dicular to it are 21 ft. and 121 ft. 16. What is the area of one of the triangles into which a diagonal of a field 10 rd. square divides it? 17. What will it cost to fence a yard in a shape of an equilateral triangle 36 ft. long with 4 lines of wire weighing one pound to every 20 ft. and costing 31 a pound? There are 12 posts, costing 15 each 1. An area bounded by a curved line, all points of which are equally distant from a point within called the center, is a circle. 2. The distance from the center to the curve is the radius, and the curve is called the circumference. In the figure how many radii are drawn? Name them. 3. Any straight line through the center terminating in the circumference is a diameter. Is a diameter shown in the figure? Which lines are diameters? A D E C B 4. Into how many equal parts does a diameter divide a circle? One half of a circle is called a semicircle. 5. Into how many equal parts do two perpendicular diameters divide a circle? Such parts are called quadrants. 6. Any part of a circumference is called an arc. Any part of a circle bounded by two radii and an arc is a sector. Name some sectors in the figure. Is a quadrant a sector? 7. For convenience in measuring arcs, every circumference, whether large or small, is divided into 360 equal parts called degrees (360°). How many degrees in a semicircumference? In a quadrant? 8. Each degree is divided into 60 minutes (60′), and each minute into 60 seconds (60'). 15° = x'. 300' = x°. An arc of 30° contains x'. of a circumference = xo. 9. Cut circles of different sizes from stiff cardboard or bring to the class several circular objects: plates, rings, covers, wheels, or coins. Measure very accurately the diameter and circumference of each. NOTE. To get an accurate measurement of a circumference two pupils can work together to an advantage. Take two rulers, one standing edgewise on the other as a guide for the circle. Mark a point on the circumference and roll through one complete revolution, noticing the distance passed over on the bottom ruler, and holding against the second ruler to get a straight path. |