EQUIVALENCES INVOLVING RECTANGLES

Rectangles of wholes and parts.

39. THEOREM 11. If there are two lines, one of which is divided into any number of parts at given points, the rectangle of the two given lines is equivalent to the sum of the rectangles of the undivided line and the several parts of the divided line.

Let AB, CF be the two lines, and let CF be divided at the points D and E into the parts CD, DE, EF.

To prove that the rectangle of AB and CF is equal to the sum of the rectangles of AB, CD; AB, DE; AB, EF.

Draw the line CG perpendicular to CF and equal to AB. Complete the rectangle CFLG, and draw DH, EK perpendicu lar to CF.

The lines DH, EK are equal to CG (I. 153) and therefore equal to AB.

The rectangle CL is equivalent to the sum of the rectangles CH, DK, EL.

Now CH is the rectangle of CG and CD, that is, of AB and CD; also DK is the rectangle of AB and DE; and EL is the rectangle of AB and EF.

Therefore the theorem is established.

NOTE. Two of the following corollaries are special cases of this theorem, and the third is an extension of it.

Rectangle of whole line and one part.

40 (a). Cor. 1. If a line is divided into any two parts, the rectangle of the whole line and one part is equivalent to the square on that part together with the rectangle of the two parts.

Square on whole line.

40 (6). Cor. 2. If a line is divided into any two parts, the square on the whole line is equivalent to the sum of the rectangles of the whole line and each of the parts.

Distributive property of rectangles.

41. Cor. 3. If each of two lines is divided into any number of parts, then the rectangle contained by the whole lines is equivalent to the sum of all the rectangles contained by each part of one and each part of the other.

[Prove by repeated applications of 39; or else by an independent figure.]

NOTE. This important principle will be referred to as "the distributive property of rectangles"; it lies at the foundation of many of the subsequent theorems.

Ex. 1. Show that 39, 40 are special cases of the "distributive property."

Ex. 2. If a line is divided into three parts, then the square on the whole line is equivalent to the sum of the rectangles of the whole line and each of its parts.

Squares on whole and parts.

42. THEOREM 12. If a line is divided into any two parts, the square on the whole line is equivalent to the sum of the squares of the parts and double the rectangle contained by the parts.

Let AB be the given line divided at E.

To prove that the square on AB is equivalent to the sum of the squares on AE and EB, and twice the rectangle of AE and EB.

[Use 36, 37,

38.]

Symbolic Proof. Another simple proof may be given by using the distributive property of rectangles. For brevity denote the rectangle of two lines AB and CD by [AB, CD], and the square on AB by the symbol sq. AB. Let the symbol stand for the phrase "is equivalent to "; the sign + for "added to" or "increased by "; and the sign for "diminished by."

Since

and

sq. AB≈[AB, AE]+[AB, EB]; [AB, AE]≈sq. AE + [AE, EB],

[40 (b)

[40 (a)

[AB, EB]≈ sq. EB + [AE, EB];

hence

sq. AB≈sq. AE + sq. EB + 2 [AE, EB].

Square on sum.

43. Cor. I. The square on the sum of two lines is equivalent to the sum of their squares and twice their rectangle.

44. Cor. 2. The square on any line is equivalent to four times the square on its half.

Ex. 1. Prove 43 by applying the distributive property to two lines each equal to the sum of the two given lines.

Ex. 2. Prove 44 by applying the distributive property to two equal lines each of which is bisected.

Ex. 3. If a line is divided into three parts, the square on the whole line is equivalent to the sum of the squares on the parts together with twice the rectangles of the parts taken two and two.

Sum of squares on whole and part.

45. THEOREM 13. If a line is divided into any two parts, the sum of the squares on the whole line and one part is equivalent to twice the rectangle of the whole line and that part, together with the square on the other part.

Let AB be the given line divided at E.

To prove that the sum of the squares of AB and EB is equivalent to twice the rectangle of AB and EB, together with the square on AE.

On AB describe a square, and complete the construction as in the figure of the preceding theorem.

The square DB is equivalent to the sum of the square DF and the rectangles HE and GB. Add to each of these equivalents the square FB. Then the sum of the squares DB and FB is equivalent to the sum of the square DF and the rectangles HB and GB. Now the latter rectangles are each equal to the rectangle of AB and EB. Hence the theorem is proved.

sq. AB+ sq. EB≈[AB, EB] + sq. AE + [AE, EB] + sq. EB,

≈[AB, EB] + sq. AE + [AB, EB],

≈sq. AE + 2[AB, EB].

[40 (a)

Square on difference.

46. Cor. The square on the difference of two lines is equivalent to the sum of their squares diminished by twice their rectangle.

Square on sum of whole and part.

47. THEOREM 14. If a line is divided into any two parts, the square on the sum of the whole line and one part is equivalent to four times the rectangle of the whole line and that part, together with the square on the other part.

Let the line AB be divided at E.

To prove that the square on the sum of AB and EB is equivalent to four times the rectangle of AB and EB, together with the square on AE.

Since sq. (AB+EB)≈ sq. AB + sq. EB +2[AB, EB];

[43

and sq. AB + sq. EB2[AB, EB]+ sq. AE;

[45

hence sq. (AB+ EB)≈4[AB, EB]+sq. AE.

48. Cor. The square on the sum of two segments exceeds the square on their difference by four times their rectangle.

Rectangles of equal parts and of unequal parts. 49. THEOREM 15. If a line is divided into two equal parts, and also into two unequal parts, the rectangle of the unequal parts, together with the square on the intermediate part, is equivalent to the square on half the line.

Let the line AB be divided into equal parts at C, and into unequal parts at D.