Εικόνες σελίδας
PDF
Ηλεκτρ. έκδοση

PROPOSITION IV. THEOREM. 372. Two regular polygons of the same number of sides are similar.

[ocr errors]

Let Q and Q be two regular polygons, each having

n sides.
We are to prove Q and Q' similar polygons.

The sum of the interior ss of each polygon is equal to 2 rt. (n 2),

§ 157 (the sum of the interior é of a polygon is equal to 2 rt. É taken as many times less 2 as the polygon has sides).

2 rt. 4 (n 2). Each Z of the polygon Q =

$ 158

n (for the of a regular polygon are all equal, and hence each Z is equal

to the sum of the És divided by their number).
Also, each 2 of Q' =

2 rt. 6 (n 2). $ 158

n
.. the two polygons Q and Q are mutually equiangular.
Moreover, AB = 1,

BC
(the sides of a regular polygon are all equal);

A' B' = 1.
and

§ 363 B'C' . A B A' B'

Ax. 1 ..BC B'C' .. the two polygons have their homologous sides proportional ; .. the two polygons are similar. $ 278

Q. E. D.

§ 363

[blocks in formation]

373. The homologous sides of similar regular polygons have the same ratio as the radii of their circumscribed circles, and, also as the radii of their inscribed circles.

[ocr errors][ocr errors][merged small][merged small][merged small][ocr errors]

Let ( and O' be the centres of the two similar regu

lar polygons A B C, etc., and A' B'C', etc.

From ( and Odraw 0 E, OD, O'E', O'D', also the Is Om and O'm'.

O E and O' E' are radii of the circumscribed ©, § 367
and 0 m and O' m' are radii of the inscribed ©. $ 368

EDO E Om
We are to prove Fudi = O' E - O'm'

In the A O E D and O' E' D' the < 0 E D, O D Е, O' E' D' and O'D' E' are equal, § 371 (being halves of the equal & FED, E DC, F' E' D' and E' D'C');

.. the A O E D and O' E' D' are similar, § 280 (if two have two ts of the one equal respectively to two ts of the other, they

are similar).
. ED OE

$ 278
". E' D = 0'E''
(the homologous sides of similar A are proportional).

E D Om
Also,

§ 297 E' D' = 0' mi' (the homologous altitudes of similar A have the same ratio as their homologous bases).

Q. E. D.

PROPOSITION VI. THEOREM.

374. The perimeters of similar regular polygons have the same ratio as the radii of their circumscribed circles, and, also as the radii of their inscribed circles.

[ocr errors][merged small][ocr errors][merged small]

Let P and Prepresent the perimeters of the two

similar regular polygons A BC, etc., and A'B'C', etc. From centres 0, O' draw 0 E, O' E', and is om and Om'. We are to prove P _ OE Om

O' E - O'm'

P ED
P = E' D''

$ 295 (the perimeters of similar polygons have the same ratio as any two homolo

gous sides).

OE ED
Moreover,
O'E' = E' DI

§ 373 (the homologous sides of similar regular polygons have the same ratio as the

radii of their circumscribed ©).

Om ED
Also
Om EDI'

§ 373

(the homologous sides of similar regular polygons have the same ratio as

the radii of their inscribed ©).
: P OE Om
r. p = 0' E = O'm''

Q. E. D.

PROPOSITION VII. THEOREM. 375. The circumferences of circles have the same ratio as their radii.

Let C and Co be the circumferences, R and R the

radii of the two circles Q and Q.
We are to prove C : C' :: R : R'.

Inscribe in the © two regular polygons of the same number of sides.

Conceive the number of the sides of these similar regular polygons to be indefinitely increased, the polygons continuing to be inscribed, and to have the same number of sides.

Then the perimeters will continue to have the same ratio as the radii of their circumscribed circles,

$ 374 (the perimeters of similar regular polygons have the same ratio as the radii

of their circumscribed ©), and will approach indefinitely to the circumferences as their limits.

.. the circumferences will have the same ratio as the radii of their circles,

§ 199 1.C : C" :: R : R'.

Q. E. D.

376. COROLLARY. By multiplying by 2, both terms of the ratio R : R', we have

C: C' :: 2 R : 2 R';

that is, the circumferences of circles are to each other as their diameters. Since

C: C' :: 2 R : 2 R',
C : 2 R :: 0 : 2 R',

§ 262

or,

That is, the ratio of the circumference of a circle to its diameter is a constant quantity.

This constant quantity is denoted by the Greek letter .

377. SCHOLIUM. The ratio is incommensurable, and therefore can be expressed only approximately in figures. The letter it, however, is used to represent its exact value.

Ex. 1. Show that two triangles which have an angle of the one equal to the supplement of the angle of the other are to each other as the products of the sides including the supplementary angles.

2. Show, geometrically, that the square described upon the sum of two straight lines is equivalent to the sum of the squares described upon the two lines plus twice their rectangle.

3. Show, geometrically, that the square described upon the difference of two straight lines is equivalent to the sum of the squares described upon the two lines minus twice their rectangle.

4. Show, geometrically, that the rectangle of the sum and difference of two straight lines is equivalent to the difference of the squares on those lines,

« ΠροηγούμενηΣυνέχεια »