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PROPOSITION XIV. THEOREM. 342. Similar triangles are to each other as the squares on their homologous sides.

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А?

ВІ
Let the two triangles be AC B and A'C' B'.

A ACB ABC

We are to prove DAC B

A Biz

Draw the perpendiculars C O and C' O'.
Than A ACB A B XCO A B co

*** A A'C' B = A' B' X C O = Ā B C Oi! 9020 (two S are to each other as the products of their bases by their altitudes).

AB co
But
A' B - GO

§ 297 (the homologous altitudes of similar A have the same ratio as their homolo

gous bases).

Substitute, in the above equality, for

tot d' o, its equal AB.

A Rii

A ACB
A A'C' B

AB
A B

AB - A B2
A' B

then

*

Q. E. D.

PROPOSITION XV. THEOREM.

343. Two similar polygons are to each other as the squares on any two homologous sides.

B

Let the two similar polygons be A B C, etc., and

A' B C', etc..
We are to prove

ABC, etc. A B

A'B'C', etc. A' Bi2
From the homologous vertices A and A' draw diagonals.

B
Now

A B BO C D

A B = B C = C Di' etc.,
(similar polygons have their homologous sides proportional);

ĀB2 B C D 2
.. by squaring,

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2 etc.

The S ABC, AC D, etc., are respectively similar to A'B'C', A' C'D', etc.,

§ 294 (two similar polygons are composed of the same number of A similar to each

other and similarly placed).

$ 342

. Δ Α Β

A B2

A A' B' C AT Bi2' (similar S are to each other as the squares on their homologous sides),

AACD C D and

§ 342 A A'C' D = CD2

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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.

: A ABC AACD A ADE A AEF " A A'B'C' A AC' D' TA A D' E AA' E' F\' . A ABC + A C D + A D E + A EF A ABC A A' B'C" + A' C' D' + 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 A B C A Bo
But
A A' B' C = A' B12

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

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

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 :: A B : AB?, (similar polygons are to each other as the squarcs of their homologous sides).

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On CONSTRUCTIONS.

PROPOSITION XVI. PROBLEM.

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

---

A---------

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
BC=A BP + 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 B C.

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

S= R + R'.'
i. 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.
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.
For

draw BC.

A B? + ACP = B T?,

§ 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 A , BC”, and A B, S, R', and R respectively; then

S= R' R.

•. S is the square required.

Q. E. F.

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