Εικόνες σελίδας
PDF
Ηλεκτρ. έκδοση

PROPOSITION XXV. PROBLEM.

357. To construct a parallelogram equivalent to a given square, and having the sum of its base and altitude equal to a given line.

R

P

M

N

Let R be the given square, and let the sum of the base and altitude of the required parallelogram be equal to the given line MN.

It is required to construct a

=

of its base and altitude MN.

=

R, and having the sum

Upon M N as a diameter, describe a semicircle. At M erect a MP, equal to a side of the given square R. to MN, cutting the circumference at S.

Draw PQ

Draw SCL to MN.

Any having CM for its altitude and CN for its base,

is equivalent to R.

For

But

SC is to PM,

(two straight lines to the same straight line are II).

[blocks in formation]

$ 65

§ 135

$307

(alet fall from any point in a circumference to the diameter is a mean proportional between the segments of the diameter).

[blocks in formation]

(the product of the means is equal to the product of the extremes).

358. SCHOLIUM. The problem is impossible when the side.

of the square is greater than one-half the line M N.

PROPOSITION XXVI. PROBLEM.

359. To construct a parallelogram equivalent to a given square, and having the difference of its base and altitude equal to a given line.

[blocks in formation]

Let R be the given square, and let the difference of the base and altitude of the required parallelogram be equal to the given line MN.

It is required to construct a of the base and altitude MN.

=

[ocr errors]

R, with the difference

Upon the given line MN as a diameter, describe a circle.

From M draw MS, tangent to the O, and equal to a side of the given square R.

Through the centre of the O, draw SB intersecting the circumference at C and B.

Then any, as R', having SB for its base and SC for its altitude, is equivalent to R.

For

SB: SM:: S M : S C,

§ 292

(if from a point without a O, a secant and a tangent be drawn, the tangent is a mean proportional between the whole secant and the part without the O).

Then

S M2

=

SBX SC;

§ 259

and the difference between SB and SC is the diameter

of the O, that is, MN.

Q. E. F.

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

Let m represent the unit of length.

It is required to find a line which shall represent the square root of 2.

On the indefinite line A B, take A C

=

and CD m,

=

= 2 m.

On A D as a diameter describe a semi-circumference.

At C erect a to A B, intersecting the circumference at E.

Then

For

CE is the line required.

ACCE:CE: CD,

§ 307

(the let fall from any point in the circumference to the diameter, is a mean proportional between the segments of the diameter);

[blocks in formation]

2. Given 2:x:: x: 3; to construct x.

3. Construct a square equivalent to a given hexagon.

PROPOSITION XXVIII. PROBLEM.

361. To construct a polygon similar to a given polygon P, and equivalent to a given polygon Q.

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

Let P and Q be two given polygons, and AB a side of polygon P.

It is required to construct a polygon similar to P and equiva

[blocks in formation]

Find a fourth proportional to m, n, and A B.

§ 304

Let this fourth proportional be A' B'.

Upon A'B', homologous to AB, construct the polygon P

similar to the given polygon P.

Then P' is the polygon required.

[blocks in formation]

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

[blocks in formation]

.. P' is equivalent to Q, and is similar to P by construction.

Q. E. F.

Ex. 1. Construct a square equivalent to the sum of three given squares whose sides are respectively 2, 3, and 5.

2. Construct a square equivalent to the difference of two given squares whose sides are respectively 7 and 3.

3. Construct a square equivalent to the sum of a given triangle and a given parallelogram.

4. Construct a rectangle having the difference of its base and altitude equal to a given line, and its area equivalent to the sum of a given triangle and a given pentagon.

5. Given a hexagon; to construct a similar hexagon whose area shall be to that of the given hexagon as 3 to 2.

6. Construct a pentagon similar to a given pentagon and equivalent to a given trapezoid.

« ΠροηγούμενηΣυνέχεια »