Ex. 1073. If two circles are tangent internally, chords of the greater circle drawn from the point of contact are divided proportionally.

*Ex. 1074. If in a triangle the squares of two unequal sides have the same ratio as their projections upon the third side, the triangle is a right triangle.

*Ex. 1075. If from a point O, OA, OB, OC, and OD are drawn so that the angle AOB is equal to the angle BOC, and the angle BOD equal to a right angle, any line intersecting OA, OB, OC, and OD is divided harmonically. (301, 302.)

* Ex. 1076. The sum of the squares of the four sides of any quadrilateral is equal to the sum of the squares of the diagonals plus four times the square of the line joining the mid-points of the diagonals. (337.)

* Ex. 1077. If from a point within the triangle ABC the perpendiculars OX, OY, and OZ be drawn upon AB, BC, and CA respectively, AX2 + BY2 + CZ2 = BX2 + YC2 + ZA2.

* Ex. 1078. State and prove the converse of the preceding exercise. *Ex. 1079. If two circles are tangent externally, either common external tangent is a mean proportional between the diame

ters.

* Ex. 1080. If in ▲ ABC point D is taken in AB such that AD: DB=1:2, and CDt, prove that

B

A

9t22 c26 b2 + 3 a2.

BOOK IV

AREAS OF POLYGONS

341. DEF. The unit of surface is a square whose side is the unit of length, etc.

Thus, a square 1 in. long and 1 in. wide is a square inch. Or a square 1 yd. long and 1 yd. wide is a square yard.

342. The area of a surface is the number of units of surface it contains.

Thus, if the floor of a room is 25 ft. long and 15 ft. wide, it contains 15 x 25 or 375 sq. ft. Hence the area of the floor is 375 sq. ft.

343. Two figures are equivalent or equal if their areas are equal.

Thus, if the area of ▲ ABC = 25 sq. ft., and the area of MNOP =25 sq. ft., then ▲ ABC is equivalent to □ MNOP, or in symbols:

=

2

NOTE. The symbol refers to the size of figures, while the symbol relates to their shape. The symbol of congruence (~) relates to both size and shape. The symbol ~ has no meaning for figures that cannot differ in shape, e.g. for straight lines. Thence the symbol of congruence () when applied to straight lines has the same significance as the symbol of equality (=).

66

If the symbol of equality (=) refers to areas, it may read either "is equivalent to or equals." Since many authors, however, designate congruent figures as equal figures, confusion may be avoided by giving preference to the term equivalent. The use of a particular symbol for equivalence () cannot be recommended.

PROPOSITION I. THEOREM

344. Rectangles having equal altitudes are to each other as their bases.

Given R and R' the areas of two rectangles whose common altitude equals h, and whose bases are respectively b and b'.

b' is divided into n equal parts, and one of these parts is laid off on b, b contains m such parts.

If perpendiculars are erected at the points of division, R is divided into m rectangles and R' is divided into n rectangles. These rectangles are all equal.

150)

(Ax. 1.)

CASE II.

is an irrational number. Since any approxi

b'

is a rational number, it must equal the corre

b! sponding approximate value of

corresponding approximate values of

b'

R

and are equal.

R'

(223)

345. COR. Rectangles having equal bases are to each other as their altitudes.

Ex. 1081. To divide a given rectangle into three equivalent parts. Ex. 1082. From a given rectangle, to cut off another rectangle whose area is of the given one.

Ex. 1083. To construct a rectangle, which is to a given one as m: n, when m and n are two given lines.

PROPOSITION II. THEOREM

346. The areas of two rectangles are to each other as the products of their bases and altitudes.

Given rectangles R and R' having the bases b and b' and the altitudes a and a' respectively.

Proof. Construct the rectangle Q, having the same base as R', and the same altitude as R.

a side of 10 ft.

Ex. 1084. Find the ratio of a rectangle 4 by 5 ft., and a square having

Ex. 1085. The diagonal of a rectangle is 26 in. long, and one of its The diagonal of another rectangle is 25 in. long, and one of Find the ratio of the areas of the two rectangles.

sides 24 in.

its sides 20 in.

347. The area of a rectangle is equal to the product of its base and altitude.

R

b

To prove

Proof.

Given R a rectangle with base b and altitude a.

R = axb.

Then

U

(346)

(342)

Q. E. D.

348. COR. The area of a square is equal to the square of its side.

349. REMARK. If the base and altitude of a rectangle are expressed by integral numbers, Prop. III may be proved by dividing the figure into squares.

Thus, if the base contains 5 and the altitude 3 linear units, the figure can be divided into fifteen squares, each being the unit of surface.

Ex. 1086. A rectangular field is 24 yd. long and 15 yd. wide. the area.

Find