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But

TD AB
C D72AB12

A ABC
A A' B' "-

A AC D
A A'C' DI

"

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.

. ΔΑΒΟ ΔΑCD ΔΑΠΕ ΔΑEF' "'AA'B'C' A AC' D - AA' D' E' DA' E'F' . A ABC + ACD + A D E + A EF A ABC

À A' B' C + A' CD' + A' D' E' + A' E F A A' B'C'' (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).

A ABC A B2
But

§ 342 A A' B' A Bi2' (similar S are to each other as the squares on their homologous sides) ;

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

A B
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. Then

S :S :: Ā Bo : A B2, (similar polygons are to each other as the squares of their homologous sides).

$ 268

V5 : VS" :: AB : A' B',
A B : A' B' :: VS : VS.

ON CONSTRUCTIONS.

PROPOSITION XVI. PROBLEM.

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

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Let R and R'be two given squares.

It is required to construct a square =

R + R'.

Construct the rt. 2 A.

Take A B equal to a side of R,
and A C equal to a side of R'.

Draw BC.
Then B C will be a side of the square required.

For
BCC= A B+ A C?,

§ 331 (the square on the hypotenuse of a rt. A 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, A B and AC", S, R, and R' respectively;

then

S = R + R.

.. S is the square required.

Q. E. F.

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347. To construct a square equivalent to the difference of two given squares.

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Let R be the smaller square and R the larger.

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

For

Construct the rt. 2 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 A C will be a side of the square required.

draw BC.
A B + A C = BC,

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

on the hypotenuse).

By transposing, À CP = B C A B. Construct the square S, having each of its sides equal to A C.

Substitute for AC, BC", and A B, S, R', and R respectively ; then

S= R' R.

.. S is the square required.

Q. E. F.

PROPOSITION XVIII. PROBLEM. 348. To construct a square equivalent to the sum of any number of given squares.

A----------

Let m, n, o, p, 7 be sides of the given squares.

It is required to construct a square = m+ n2 + 02 + p + ge.
Take A B = m.
Draw AC=n and I to A B at A.

Draw BC.
Draw C E= 0 and I to B C at C, and draw B E.
Draw E F=p and I to B E at E, and draw B F.
Draw FH=r and I to B Fat F, and draw B H.
The square constructed on B H is the square required.
For BH = FHP + BF,

= FH + EF? + E B,
= F 11+ E F + E C + C B,

= FHP + Ē F + E C + C A + A B, $ 331 (the sum of the squares on two sides of a rt. A is equivalent to the square

on the hypotenuse). Substitute for A B, CA, EC, EF, and FH, m, n, 0, P, and r respectively;

then BTI= m2 + m2 + 02 + p2 + pod.

Q. E. F.

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

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A B A BI

---

P ------------Let R and R' be two similar polygons, and A B and

A' B' two homologous sides.

It is required to construct a similar polygon equivalent to R + R'. Construct the rt. 2 P. Take PH= A' B', and PO=A B.

Draw 0 H.

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 B C : A B,

§ 343 (similar polygons are to each other as the squares on their homologous sides). Also R" : R' : : A" BM : AB-.

§ 343 In the first proportion, by composition, R' + R : R' :: ABS + Ā B : A Bio, $ 264

:: PHP + PO? PT',

:: H O : PH.
But R" : R' :: A" B12 : A' B,

:: | : PH.
..R" : R' :: R' + R : R';
.. R' = R' + R.

Q. E. F.

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