THEOREM 6. The parallelogram formed by joining the middle points of the consecutive sides of a quadrilateral has half the area of the quadrilateral (§ 136).

THEOREM 7. If through the middle point of one of the non-parallel sides of a trapezoid we draw a line parallel to the opposite side, and complete the parallelogram, the area of the parallelogram will be equal to that of the trapezoid.

THEOREM 8. If we join the middle of one of the nonparallel sides of a trapezoid to the ends of the opposite side, the middle triangle will have half the area of the trapezoid.

THEOREM 9. If two triangles have two sides of the one equal to two sides

of the other respectively, and the included angles supplemen

tary, they are

equal in area.

M

Hypothesis. CA = MK. CB= ML.

Angle ACB+ angle KML: 180°.

Conclusion. Area ABC area KLM.

=

THEOREM 10. The sum of the squares upon the diagonals

of a trapezoid are equal to

D

C

Conclusion. AC2 + BD2 = AD2 + BC2 + 2AB. CD.

THEOREM 11. If from any point within a polygon perpendiculars be dropped upon the sides, the sum of the squares of one set of alternate segments is equal to the sum of the squares of the other set.

E

d

D

с

B

B

Aa2 + Bb2 + Сc2 + Dd2 + Ee2 = aB2 + bC2 + cD2 + dE2 + eA2.

NUMERICAL EXERCISES.

1. In a right-angled triangle the lengths of the sides containing the right angle are 9 and 12 feet. What is the length of the hypothenuse? What is the area of the triangle?

2. If the length of the hypothenuse is 10 feet, and that of one side 8 feet, what is the length of the remaining side? What is the area of the triangle?

3. In a right-angled triangle the perpendicular from the right angle upon the hypothenuse divides the latter into segments which are respectively 9 and 16 feet. Find the lengths of the perpendicular and of the two sides, and the area of the triangle.

4. What three different expressions for the area of a triangle may we obtain from § 301 by taking different sides as the base? What theorem hence follows?

5. What is the area of the triangle of which the respective sides are 15, 41, and 52 metres?

6. If the diagonal of a rectangle is 13 feet, and one of the sides 12 feet, what is the area?

7. Show how the altitude and area of a trapezoid may be computed when its four sides are known.

Refer to the computation of the altitude p of a triangle in § 333. 8. If each side of an equilateral triangle is unity, find its altitude.

9. Draw an equilateral triangle, ABC. Show that the bisectors of each interior angle will bisect the opposite side perpendicularly. Show that if the bisector of C be produced beyond the point O in which it meets the other bisectors and intersect the opposite side in D, and if we take DF= DO and join AF, BF, then—

A

B

¡D

I. OAF and OBF will be equilateral triangles.
II. The points A, C, B, F lie on a circle.

III. The lines OA, OB, and OC will all be equal.

Also, supposing the length of each side of the triangle ABC to be unity, compute the lengths of OC and OD.

BOOK V.

THE PROPORTION OF MAGNITUDES.

CHAPTER I.

RATIO AND PROPORTION OF MAGNITUDES IN

GENERAL.

334. Definition. When a greater magnitude contains a lesser one an exact number of times, the greater one is said to be a multiple of the lesser, and the lesser is said to measure the greater, and to be an aliquot part of the greater.

335. Def. When a lesser magnitude can be found which is a measure of each of two greater ones, the latter are said to be commensurable, and the former is said to be a common measure of them.

336. Def. When two magnitudes have no common measure they are said to be incommensurable.

337. Def. When one of two commensurable magnitudes contains the common measure m times, and the other contains it n times, they are said to be to each other as m to n.

B as 5 to 3, and a is a common measure of A and B.

EXERCISES. Draw, by the eye, pairs of lines which shall

be to each other as 3 to 4; as 2 to 5; as 4 to 7; as 5 to 6; as 7 to 2.

338. Corollary. If A is to B as m to n, then, by definition (§ 337), the mth part of A will be equal to the nth part of B; or, in symbolic language,

NOTATION. The statement that two magnitudes A and B are to each other as the numbers m and n is written symbolically

or

A: B: m : n,

A: Bm : n.

NOTE. The second form, or that of an equation, is preferable, and is most used by mathematicians; but the first form is more common in elementary books.

339. Def. If a pair of magnitudes A and B are to each other as two numbers m and n, and another pair P and Q are also to each other as mn, then we say that A is to B as P to Q, and the four magnitudes A, B, P, and Q are said to be proportional or to form a proportion.

NOTATION. The statement that the four magnitudes A, B, P, and Q are proportional is expressed in the symbolic form A B P 2,

or

A BP Q;

which is read: A is to B as P is to Q.

340. Def. The symbolic statement that four magnitudes are proportional

is called a proportion.

α

α

A

341. Def. The four quantities which form a proportion are called terms of the proportion.

342. Def. The first and fourth terms of a proportion are called the extremes; the second and third, the means.

EXAMPLE. In the last proportion A and Q are the extremes, B and P the means.

343. Def. The first and third terms, which precede the symbol, are called antecedents; the second and fourth, which follow the symbol:, are called consequents.

EXAMPLE. In the last proportion A and P are the antecedents, B and Q the consequents.

344. Def. If the means are equal, each of them is said to be a mean proportional between the extremes, and the three quantities are said to be in proportion.

Axioms.

345. Ax. 1. If there be a greater and a lesser magnitude of the same kind, the greater may be divided into so many equal parts that each part shall be less than the lesser magnitude.

NOTE. By magnitudes of the same kind are meant those which are both numbers, both lines, both surfaces, or both solids.

Ax. 2. If a greater magnitude be a certain number of times a lesser one, then any multiple of that greater one will be the same number of times the corresponding multiple of the lesser.

then

Symbolic expression of this axiom. If mag. Gix mag. L, nG=ix nL.

Ax. 3. If a lesser magnitude be a certain aliquot part of a greater one, then any multiple of the lesser one will be the same aliquot part of the corresponding multiple of the greater.

Symbolic expression of this axiom. If mag. L=

mag. G