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PROPOSITION XIV. THEOREM. 342. Similar triangles are to each other as the squares on their homologous sides.
A ACB ABC
We are to prove DAC B
Draw the perpendiculars C O and C' O'.
*** 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).
§ 297 (the homologous altitudes of similar A have the same ratio as their homolo
Substitute, in the above equality, for
tot d' o, its equal AB.
AB - A B2
Q. E. D.
PROPOSITION XV. THEOREM.
343. Two similar polygons are to each other as the squares on any two homologous sides.
Let the two similar polygons be A B C, etc., and
A' B C', etc..
ABC, etc. A B
A'B'C', etc. A' Bi2
A B BO C D
A B = B C = C Di' etc.,
ĀB2 B C D 2
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).
. Δ Α Β
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
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
§ 342 (similar & are to each other as the squares on their homologous sides) ;
the polygon A B C, etc. A B
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).
PROPOSITION XVI. PROBLEM.
346. To construct a square equivalent to the sum of two given squares.
Let R and R' be two given squares.
It is required to construct a square = R + R'.
Take A B equal to a side of R,
: Draw BC.
§ 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'.'
Q. E. F.
347. To construct a square equivalent to the difference of two given squares.
Let R be the smaller square and R the larger.
It is required to construct a square = R' – R.
Take A B 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.
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 C°, BC”, and A B”, S, R', and R respectively; then
S= R' – R.
•. S is the square required.
Q. E. F.