PROPOSITION I. THEOREM 327. Two rectangles having equal altitudes are to each other as their bases. Hypothesis. Rectangles ABCD and EFGH have equal altitudes AB and EF, and bases AD and EH, respectively. CASE I. Assume that AD and EH are commensurable. § 211 Proof. 1. Let AK, a common measure of AD and EH, be contained in AD 5 times and in EH 3 times. Draw Is to AD and EH at the points of division. 2. Then ABCD and EFGH are divided into equal rectangles. (Complete the proof.) Why? Suggestions. What is the value of ? of EH ? Then compare these EFGH AD ABCD ratios. CASE II. When AD and EH are incommensurable, the theorem is still true. The proof is given in § 425. 328. Cor. Two rectangles having equal bases are to each other as their altitudes. Ex. 7. Construct a rectangle which will be three times a given rectangle; also one which will be three fourths a given rectangle. Ex. 8. Two rectangles M and T have equal bases b and altitudes r and s respectively. What is the ratio of M to T'? Note. Supplementary Exercises 4 to 5, p. 295, can be studied now. PROPOSITION II. THEOREM 329. Two rectangles are to each other as the products of their bases by their altitudes. a M N R b b' Hypothesis. Rectangle M has base b and altitude a; rectangle N has base b' and altitude a'. Proof. 1. Let rectangle R have base b' and altitude a. since M and R have equal altitudes. Why? = R Ex. 9. M Ꭱ ab Ꭱ N a'b' N What is the ratio of rectangles R and S if their dimensions are as follows? M ab or = a'b' Why? PROPOSITION III. THEOREM 330. The area of a rectangle is the product of its base and altitude. Hypothesis. Rectangle M has altitude a and base b. Proof. 1. Let square N be the unit of surface measure. 2. = § 320 Why? 4. Note. Remember that this theorem means that the number of square units in the area equals the product of the number of linear units in the base by the number in the altitude. A similar interpretation must be given for each of the measurement theorems of this Book. Ex. 11. A business corner 50 ft. x 120 ft. is valued at $9000. What is the value per square foot ? Ex. 12. The area of a square is 590.49 sq. ft. Find its perimeter. Ex. 13. A rectangle has the dimensions 30 ft. and 120 ft. Compare its perimeter with that of an equal square. Ex. 14. An ordinary eight-room house costs approximately $4.75 per square foot of ground covered by it. What is the approximate cost of a house 27 ft. x 36 ft.? Ex. 15. The area of a rectangle is 147 sq. ft. Its base is three times its altitude. What are its dimensions ? Ex. 16. What are the dimensions of a rectangle whose area is 168 sq. ft. and whose perimeter is 58 ft ? Suggestion. Let the base = x and the altitude and complete the solution algebraically. = y. Form two equations Ex. 17. What is the length of the diagonal of a rectangle whose area is 2640 sq. ft., if its altitude is 48 ft.? PROPOSITION IV. THEOREM 331. The area of a parallelogram equals the product of its base and altitude. Hypothesis. ABCD is a parallelogram. Proof. 1. Draw AE || DF, meeting BC extended at E. .. AEFD is a rectangle. 2. Why? Give the full proof. Why? Why? Ax. 1, § 51 332. Corollaries. Let P1 have base b1 and altitude a1; P, have base b2 and altitude a2. and (1) Parallelograms having equal bases and equal altitudes are equal. (2) Two parallelograms are to each other as the products of their bases by their altitudes. For, since P1 = a1b1 and □ P2 = a2b2, then (3) Parallelograms having equal altitudes are to each other as their bases. For, in (2), if a1 = a2, then □ P1: P2 = b1: b2. (4) Parallelograms having equal bases are to each other as their altitudes. Ex. 18. (a) (b) (c) Ex. 19. What is the area of R, of S, and of ☐ T? R has altitude 4 in. and base 9 in. S has altitude 15 ft. and base 20 ft. T has altitude 3 x in. and base 11 y in. What is the altitude of a parallelogram whose area is 56 sq. in., if its base is 14 in. ? Ex. 20. gram. Construct a parallelogram equal to twice a given parallelo Ex. 21. Construct a rectangle equal to two thirds a given parallelogram. Ex. 22. Divide a parallelogram into two equal parallelograms; into four equal parallelograms. Ex. 23. What is the ratio of P to R if the base of each is 10 in. and the altitudes are 5 in. and 8 in. respectively? 3 in. and BC = 4 in., Ex. 24. Construct a ABCD having AB and having: (a) ≤ B = 30°; (b) ZB = 45°. (c) Determine the area of each of the parallelograms. Ex. 25. The base of A ABC is 10 and the altitude is 5. What is the area of ▲ ABC? Suggestion. - Draw AD || BC and CD || AB to form ABCD. Compare ABC with ABCD. Then determine the area of □ ABCD and finally of ▲ ABC. B E PROPOSITION V. THEOREM 5 10 333. The area of a triangle equals one half the product of its base and altitude. Hypothesis. A ABC has altitude AE =a and base BC= b. Conclusion. Area of ▲ ABC = 1⁄2 ab. [Proof to be given by the pupil.] Suggestion.-Construct ABCD and proceed as in Ex. 25. |