2. What would be the length of the base of the new figure, if the distance from A to E is 8 in. and the distance from B to F is 12 in.? Why? 3. What figure would be formed if the shaded portions in this trapezoid were fitted into the spaces marked by the dotted lines? 4. Call the length of GK 6 in. and HJ 12 in. What then is the base of the equi valent figure? How is it found? 5. If the altitude of the trapezoid GKJH is 5 in.; the distance from G to K, 6 in.; and from H to J, 12 in., what is the area of the trapezoid? To find the area of a trapezoid, multiply one half the sum of the parallel sides by the altitude. 6. The parallel sides of a trapezoid are respectively 22 in. and 26 in. long. Its altitude is 10 in. What is its area in square inches? In square feet? 7. A piece of leather is a trapezoid with its parallel sides 18 in. and 24 in. long and an altitude of 16 in. At 15¢ a square foot, what is the price of the piece? 8. Building lot No. 1, pictured on the map, page 103, is a trapezoid with its parallel sides 90 ft. and 40 ft. long. The altitude of the trapezoid is 100 ft. What then is the area of the building lot? 9. Lot 7, pictured on the same map, is a trapezoid with its parallel sides 80 ft. and 30 ft. long. The altitude of the trapezoid is 110 ft. Estimate, then find, the difference in the area of these two lots. (See problem 8.) 10. Find how many acres there are in a field that is a trapezoid with parallel sides 30 rd. and 50 rd. long. Altitude of the trapezoid, 20 rd. *11. A field is in the shape of a trapezoid with parallel sides 350 ft. and 310 ft. long, with an altitude of 10 rd. What is the value of the field at $110 an acre? 47. Triangles 1. With the help of the definition of a quadrilateral on page 102, define a triangle. 2. Thinking of the form of the word triangle (tri means three), how else might a triangle be defined? Right-Angled A triangle with one of its angles a right angle is called a right-angled triangle, or a right triangle. A triangle with its three sides equal in length is called an equilateral triangle. Isosceles Triangles Equilateral A triangle with only two of its sides equal is called an isosceles triangle. The altitude of a triangle is measured by a line drawn at right angles to one of the sides taken as the base. (See dotted lines in drawings.) The point of the angle opposite the line taken as the base is called the apex of the triangle. 3. Draw an equilateral triangle, an isosceles triangle, and a rightangled triangle, indicating with a dotted line the altitude of each. 4. Draw three triangles, no one of which is right-angled, équilateral, or isosceles. Indicate the altitude of these triangles. II To draw an equilateral triangle with a compass, place the pin end of the compass at one end of the line drawn for the base of the triangle; spread the points of the compass so that the distance between them equals the length of the line drawn for the base, and draw an arc as indicated in the diagram. Next, change the pin end of the compass to the opposite end of the base line and draw another arc. The point of the intersection of the two arcs is the apex of the triangle. Connect this point with the two ends of the base line, and the result is an equilateral triangle. Drawing an 1. Draw an equilateral triangle measuring 2 inches on a side. Draw one measuring 11⁄2 inches on a side. 2. Draw a triangle with sides measuring 3, 4, and 5 inches respectively. What kind of triangle is the result? 3. Draw a triangle with sides measuring 6, 8, and 10 inches. Try other multiples of 3, 4, and 5. What kind of triangle is the result in each case? *4. With the shortest side of each not less than 2 inches in length, draw an equilateral triangle, an isosceles triangle, and a rightangled triangle. Cut them out, and then, with the help of the protractor on page 98, make careful measurements of the degrees in each of the angles. Find the sum of the angles in each triangle. D B III. THE AREA OF A TRIANGLE [Use pencil only when needed.] 1. Show by paper cutting that the triangle E ABC is equal in area to the rectangle BDEC. 2. What is the relation of the altitude of the c rectangle formed to the altitude of the triangle? 3. Show by paper cutting that the triangle FGH is equal in area to the rectangle FIJH. G H 4. Compare the base of the triangle with that of the rectangle. Compare their altitudes. 5. What rectangle is equivalent to a triangle with a base of 6 in. and altitude 10 in.? What is the area of the rectangle? Of the triangle? 6. Show that this rule is true: To find the area of a triangle, find one half the product of the base and the altitude. 7. What is the area of a triangle with a base 8 in. long and an altitude of 12 in.? With a base 41⁄2 in. long and an altitude of 6 in.? With a base of 12 ft. and an altitude of 2 yd.? 8. A sail, equivalent to a triangle with a base of 9 ft. and an altitude of 12 ft., exposes how many square feet of surface to the wind when unfurled? 9. A pennant, equivalent to a triangle with a base of 1 yd. and an altitude of 7 ft., contains how many square yards of material? 10. A triangular field with a base of 40 rd. and an altitude of 20 rd. contains how many acres? *11. Find the number of acres in a triangular lot with a base of 40 rd. and an altitude of 330 ft. *12. A certain field is a parallelogram with a base of 40 rd. and an altitude of 20 rd. This field contains how many more acres than a triangular field with the same base and altitude? *13. The gable of a house has a base of 10 ft. 9 in. and an altitude of 6 ft. 7 in. How many square feet does it contain? *14. Cut out triangles of different shapes and figure out their areas. Cakes of ice, blocks of stone, and heavy timbers are usually cut so that they have rectangular surfaces. All such solids are called rectangular prisms. In measuring the contents or volume of a prism, such cubic units as the cubic inch, the cubic foot, and the cubic yard are used. Name a use for each of these measures. In finding the number of cubic units in a prism, the length may be taken to indicate the number of cubic units that are contained in one row of one layer of the prism. The breadth then indicates the number of rows of cubic units in each layer, and the thickness indicates the number of layers of cubic units contained in the prism. For example, a rectangular prism 6 in. by 4 in. by 3 in. contains 6 cu. in. multiplied by 4 and then by 3. To express briefly directions for finding the number of cubic units in a solid, this rule is given: To find the cubical contents, or volume, of a prism, multiply the length by the breadth by the thickness. |