In like manner we may prove that the ratio of any two of the similar A is the same as that of any other two.

AA'B'C' + A' C' D' + A' D' E' + A'E' F' Δ Α' Β' Γ'

(in a series of equal ratios the sum of the antecedents is to the sum of the consequents as any antecedent is to its consequent).

(similar are to each other as the squares on their homologous sides);

the polygon ABC, etc. the polygon A'B'C', etc.

A B2

A' Bi2

Q. E. D.

344. COROLLARY 1. Similar polygons are to each other as the squares on any two homologous lines.

345. COR. 2. The homologous sides of two similar polygons have the same ratio as the square roots of their areas.

Let S and S' represent the areas of the two similar polygons A B C, etc., and A' B' C', etc., respectively.

(similar polygons are to each other as the squares of their homologous sides).

VS: √S: AB : A' B',

or,

AB: A'B': √S: √S'.

$268

ON CONSTRUCTIONS.

PROPOSITION XVI. PROBLEM.

346. To construct a square equivalent to the sum of two

given squares.

B

R'

R

S

Let R and R' be two given squares.

It is required to construct a square = R+R'.

Construct the rt. A.

For

Take A B equal to a side of R,

and AC equal to a side of R'.

Draw BC.

Then BC will be a side of the square required.

BC2 = A B2 + AC2,

§ 331

(the square on the hypotenuse of a rt. ▲ is equivalent to the sum of the

squares on the two sides).

Construct the square S, having each of its sides equal to BC.

Substitute for BC, AB and AC, S, R, and R' respectively;

then

S = R+R'.

.. S is the square required.

Q. E. F.

PROPOSITION XVII PROBLEM.

347. To construct a square equivalent to the difference

Let R be the smaller square and R' the larger.

It is required to construct a square

=

= R' — R.

Construct the rt. ▲ A.

Take A B equal to a side of R.

From B as a centre, with a radius equal to a side of R',
describe an arc cutting the line A X at C.

Then AC will be a side of the square required.

(the sum of the squares on the two sides of a rt. A is equivalent to the square

Construct the square S, having each of its sides equal to A C.

Substitute for AC2, BC2, and A B2, S, R', and R re

spectively;

then

SR' - R.

..S is the square required.

Q. E. F.

348. To construct a square equivalent to the sum of any

Let m, n, o, P, r be sides of the given squares.

It is required to construct a square

=

m2 + n2 + o2 + p2 + ‚‚2.

Draw A C n and to AB at A.

Draw BC.

Draw CE o and 1 to BC at C, and draw B E.
Draw E Fp and I to B E at E, and draw B F.
Draw FH = =r and to BF at F, and draw B H.

The square constructed on BH is the square required.
BH2 = FH2 + BF2,

For

=

F H2 + E F2 + E C2 + C B2,

= F H2 + E F2 + E C2 + C A2 + AB2, § 331

=

(the sum of the squares on two sides of a rt. ▲ is equivalent to the square

on the hypotenuse).

Substitute for AB, CA, EC, EF, and FH, m, n, o, p, andr respectively;

then

BH2

=

m2 + n2 + o2 + 202 + p2.

Q. E. F.

349. To construct a polygon similar to two given similar polygons and equivalent to their sum.

Let R and R' be two similar polygons, and AB and A'B' two homologous sides.

It is required to construct a similar polygon equivalent to R+R'.

Construct the rt. P.

Take PHA' B', and P0 = A B.

Draw OH.

Take A" B" =0 H.

Upon A" B", homologous to A B, construct the polygon R" similar to R.

Then R" is the polygon required.

For

R' : R : : A' B12 : A B2,

§ 343

(similar polygons are to each other as the squares on their homologous sides).