587. SCHOLIUM. The meaning to be attached to formula (1) is, that the number of surface units in a rectangle is the same as the product of the number of linear units in the base by the number of linear units in the altitude. If the base is 4 ft. and the altitude 3 ft., the number of square feet (surface units) in the rectangle is 4 x 3 or 12. The area then is 12 square feet. 588. COROLLARY I. The area of any parallelogram is equal to the product of its base and altitude. Let ABCD be any parallelogram and DE be its altitude. To Prove ABCD = AD X DE. B E 589. COROLLARY II. Any two parallelograms are to each other as the products of their bases and altitudes; if their bases are equal the parallelograms are to each other as their altitudes; if the altitudes are equal the parallelograms are to each other as their bases. 590. EXERCISE. Construct a square equivalent to a given parallelo gram. 591. EXERCISE. Construct a rectangle having a given base and equivalent to a given parallelogram. 592. EXERCISE. Of all equivalent parallelograms having a common base, the rectangle has the least perimeter. Of all equivalent rectangles, the square has the least perimeter. 593. The area of a triangle is one half the product of its base and altitude. E C Let ABC be any ▲, and BD its altitude. Proof. Construct the parallelogram ACBE. ACBE = AC × BD. (?) ▲ ABC=AC × BD. (?) 594. COROLLARY I. Q.E.D. Triangles are to each other as the products of their bases and altitudes; if their bases are equal the triangles are to each other as their altitudes; if their altitudes are equal the triangles are to each other as their bases. 595. COROLLARY II. The area of a triangle is one half the product of its perimeter and the radius of the inscribed circle. of A BOC and AOC. Call the radius of the inscribed circle r. the inscribed circle of a triangle is equal to the area of the triangle divided by one half its perimeter. 597. EXERCISE. The area of a rhombus is equal to one half the product of its diagonals. 598. EXERCISE. 599. EXERCISE. 600. EXERCISE. Construct a square equivalent to a given triangle. Construct a square equivalent to a given polygon. Two triangles having a common base are to each other as the segments into which the line joining their vertices is divided by the common base, or base produced. [The ABC and ACD have the com mon base AC; to prove ΔΑΒΕ BE B NOTE. When the two triangles are on the same side of the common base, BD, the line joining their vertices is divided externally at E. 601. DEFINITION. Lines that pass through a common point are called concurrent lines. 602. EXERCISE. If three concurrent lines AO, BO, and CO, drawn from the vertices of the triangle ABC, meet the opposite sides in the points D, E, B D E CE A BOC = (?) = (?) DC ▲ AOC [Draw AD and CF. Call their point of intersection 0. Draw BO. Suppose BO prolonged does not go to E, but some other point of AC, as E'. B = 1. (?) E'A EA DC EA FB CE CE = Show that this last proportion is absurd. .. AD, BE, and CF are concurrent.] 603. EXERCISE. Show by means of the converse of the last exercise that the following lines in a triangle are concurrent. 1. The medial lines. 2. The bisectors of the angles. 3. The altitudes. PROPOSITION VII. THEOREM 604. The area of a trapezoid is one half the product of its altitude and the sum of its parallel sides. Let ABCD be a trapezoid, and mn be its altitude. To Prove ABCD = mn (BC + AD). Proof. Draw the diagonal AC. Show that mn is equal to the altitude of each triangle 605. COROLLARY. The area of a trapezoid is equal to the product of the altitude and the line joining the middle points of the non-parallel sides. EF.] [EF = } (BC + AD) (?) .. ABCD = mn 606. EXERCISE. In the figure for § 604 let BC = 8 in., AD and mn = EF. Find the area of the trapezoid. 607. EXERCISE. = 12 in., Construct a square equivalent to a given trapezoid. 608. EXERCISE. Construct a rectangle equivalent to a given trapezoid and having its altitude equal to that of the trapezoid. 609. EXERCISE. The triangle formed by joining the middle point of one of the non-parallel sides of a trapezoid with the extremities of the opposite side is equivalent to one half the trapezoid. 610. EXERCISE. A straight line joining the middle points of the parallel sides of a trapezoid divides it into two equivalent figures. SANDERS' GEOM. - 13 |