THEOREM 24. [Euclid I. 35.] Parallelograms on the same base and between the same parallels are equal in area. Let the parm ABCD, EBCF be on the same base BC, and between the same par's BC, AF. It is required to prove that the parm ABCD = the parm EBCF in area. Now, if from the whole fig. ABCF the FDC is taken, the remainder is the parm ABCD. And if from the whole fig. ABCF the EAB is taken, the remainder is the par EBCF. .. these remainders are equal; that is, the parm ABCD = the parTM EBCF. Q.E.D. EXERCISE. In the above diagram the sides AD, EF overlap. Draw diagrams in which (i) these sides do not overlap; (ii) the ends E and D coincide. Go through the proof with these diagrams, and ascertain if it applies to them without change. THE AREA OF A PARALLELOGRAM. Let ABCD be a parallelogram, and ABEF the rectangle on the same base AB and of the same altitude BE. Then by Theorem 24, area of parm ABCD = area of rect. ABEF = AB X BE F D E Α ́ B COROLLARY. Since the area of a parallelogram depends only on its base and altitude, it follows that Parallelograms on equal bases and of equal altitudes are equal in area. EXERCISES. (Numerical and Graphical.) Find the area of parallelograms in which (i) the base 5.5 cm., and the height=4 cm. 2. Draw a parallelogram ABCD having given AB=21", AD=11", and the LA=65°. Draw and measure the perpendicular from D on AB, and hence calculate the approximate area. Why approximate? Again calculate the area from the length of AD and the perpendicular on it from B. Obtain the average of the two results. 3. Two adjacent sides of a parallelogram are 30 metres and 25 metres, and the included angle is 50°. Draw a plan, 1 cm. representing 5 metres; and by measuring each altitude, make two independent calculations of the area. Give the average result. 4. The area of a parallelogram ABCD is 4.2 sq. in., and the base AB is 2.8". Find the height. If AD=2", draw the parallelogram. 5. Each side of a rhombus is 2", and its area is 3.86 sq. in. Calculate an altitude. Hence draw the rhombus, and measure one of its acute angles. THEOREM 25. The Area of a Triangle. The area of a triangle is half the area of the rectangle on the same base and having the same altitude. Let ABC be a triangle, and BDEC a rectangle on the same base BC and with the same altitude AF. It is required to prove that the ABC is half the rectangle BDEC. Proof. Since AF is perp. to BC, each of the figures DF, EF is a rectangle. Because the diagonal AB bisects the rectangle DF, .. the ABF is half the rectangle DF. Similarly, the AFC is half the rectangle FE. .. adding these results in Fig. 1, and taking the difference in Fig. 2, the ABC is half the rectangle BDEC. Q.E.D. COROLLARY. A triangle is half any parallelogram on the same base and between the same parallels. For the AABC is half the rect. BCED. And the rect. BCED = any parm BCHG on the same base and between the same parls. .. the ABC is half the parm BCHG. THE AREA OF A TRIANGLE. If BC and AF respectively contain a units and p units of length, the rectangle BDEC contains ap units of area. .. the area of the ABC = ap units of area. This result may be stated thus: Area of a Triangle. base x altitude. (iii) the base = 160 metres, the height=125 metres. In each case draw and 2. Draw triangles from the following data. measure the altitude with reference to a given side as base: hence calculate the approximate area. (i) a=84 cm., b=6.8 cm., c=4.0 cm. 3. ABC is a triangle right-angled at C; shew that its area = BC × CA. Given a=6 cm., b=5 cm., calculate the area. Draw the triangle and measure the hypotenuse c; draw and measure the perpendicular from C on the hypotenuse; hence calculate the approximate area. Note the error in your approximate result, and express it as a percentage of the true value. 4. Repeat the whole process of the last question for a right-angled triangle ABC, in which a=2.8′′ and b=4.5′′; C being the right angle as before. 5. In a triangle, given (i) Area-80 sq. in., base = 1 ft. 8 in.; calculate the altitude. (ii) Area = 10.4 sq. cm., altitude 16 cm.; calculate the base. 6. Construct a triangle ABC, having given a=3′0′′, b=2·8′′, c=2·6′′. Draw and measure the perpendicular from A on BC; hence calculate the approximate area. THEOREM 26. [Euclid I. 37.] Triangles on the same base and between the same parallels (hence, of the same altitude) are equal in Proof. If BCED is the rectangle on the base BC, and between the same parallels as the given triangles, the AABC is half the rect. BCED; also the AGBC is half the rect. BCED; .. the ABC= the ▲ GBC. Theor. 25 Q.E.D. Similarly, triangles on equal bases and of equal altitudes are equal in area. し THEOREM 27. [Euclid I. 39.] If two triangles are equal in area, and stand on the same base and on the same side of it, they are between the same parallels. Let the ABC, GBC, standing on the same base BC, be equal in area; and let AF and GH be their altitudes. It is required to prove that AG and BC are par1. A G B F C H Proof. The ABC is half the rectangle contained by BC and AF; and the GBC is half the rectangle contained by BC and GH; .. the rect. BC, AF the rect. BC, GH; = AF = GH. Theor. 23, Cor. 2. Also AF and GH are par1; Q.E.D. |