The altitude of a parallelogram is its height from its base. 2. Cut into two pieces and arrange the pieces to form a rectangle. 3. Compare the bases of the parallelogram and the rectangle. 4. Compare their altitudes. 5. What is the area of the rectangle? Of the parallelogram? 6. What must be known to find the area of a parallelogram? The area of a parallelogram is equal to the product of its base and its altitude. Find the areas of parallelograms of these dimensions: 19. Make up problems in finding areas of parallelograms. FINDING AREAS OF TRAPEZOIDS The altitude of a trapezoid is the perpendicular distance between its parallel sides. 1. What is the altitude of figure 1? Of figure 2? 2. Cut from paper a trapezoid like figure 1. 3. Fold so that the upper edge is on a line with the lower edge. Crease. Cut on the line of the crease. Arrange the two pieces to form a rectangle. 4. What two sides of the trapezoid form the base of the rectangle? What is the length of the base? 5. What is the altitude of the trapezoid? The altitude of the trapezoid is what part of the altitude of the rectangle? 6. What is the area of the rectangle? Of the trapezoid? 7. Cut from paper a trapezoid like figure 2. Fold, cut, and arrange the pieces to form a parallelogram. Its alti 8. What is the base of the parallelogram? tude? Its area? What is the area of the trapezoid? A trapezoid is equal to a parallelogram whose base is the sum of the parallel sides of the trapezoid, and whose altitude is one half the altitude of the trapezoid. The area of a trapezoid is equal to one half the product of its altitude and the sum of its parallel sides. 9. What is the area of a trapezoid whose parallel sides are 24 feet and 16 feet, and whose altitude is 9 feet? 24+ 1640, sum of parallel sides. 18. What is the area of a trapezoid if the base is 20 in., the side parallel to the base 12 in., and the altitude 8 in.? 19. The two parallel sides of a trapezoid are 16 ft. and 22 ft., and the altitude is 8 ft. Find the area. 20. How many square rods in a four-sided field whose two parallel sides are, respectively, 40 rods and 30 rods, and the distance between them is 15 rods? 21. Make a problem about a trapezoid. Draw the diagram, work out the problem, and ask your teacher to give the problem to the class to solve. TRIANGLES A polygon of three sides is a triangle. I. With respect to their sides. A triangle having three equal sides is an equilateral triangle. (Fig. 1.) A triangle having two equal sides is an isosceles triangle. (Fig. 2.) A triangle having no two sides equal is a scalene tri angle. (Fig. 3.) FIGURE FIGURE 2 FIGURE 3 II. With respect to their angles. A triangle having a right angle is a right-angled triangle. (Fig. 4.) A triangle having three acute angles is an acute-angled triangle. (Fig. 5.) A triangle having an obtuse angle is an obtuse-angled triangle. (Fig. 6.) FIGURE 4 FIGURE 5 FIGURE 6 The altitude of a triangle is the perpendicular distance from the angle opposite the base to the base or to the base extended. What is the altitude of figure 4? figure 5? Of figure 6? Of 1. The perimeter of an equilateral triangle is 48 inches. What is the length of one side? 2. Each of the equal sides of an isosceles triangle is 15 inches and the base is 8 inches. What is the perimeter? 3. The perimeter of a right-angled triangle is 6 inches. The base is 11 inches and the altitude 2 inches. What is the length of the third side? 4. The perimeter of an isosceles triangle is 15 inches. The base is 3 inches. How long is each of the other sides? 5. The sides of a scalene triangle are 11⁄2 inches, 22 inches, and 3 inches. What is its perimeter? 6. A watch charm in the shape of an equilateral triangle requires a gold band 24 inches long to inclose it. What is the length of one side of the charm? 7. How many feet of wire fencing will inclose an equilateral flower bed, one of whose sides is 4.5 feet? FINDING AREAS OF TRIANGLES A straight line connecting the opposite corners of a quadrilateral is a diagonal. The line AB is a diagonal of figure 1. B FIGURE 1 1. Cut from paper a rectangle whose base is 3 inches and whose altitude is 2 inches. 2. What is the area of this rectangle? 3. Cut along the diagonal. Compare the areas of the two triangles by placing one over the other. 4. The area of each triangle is what part of the area of the rectangle? What is the area of each triangle? |