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PROPOSITION XXVIII. PROBLEM.

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

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т А В Let P and Q be two given polygons, and A B a side

of polygon P.

It is required to construct a polygon similar to P and equivalent to Q. Find a square equivalent to P,

$ 356

and let m be equal to one of its sides. Find a square equivalent to Q,

§ 356 and let n be equal to one of its sides. Find a fourth proportional to m, n, and A B.

§ 304 Let this fourth proportional be A' B'.

Upon A' B', homologous to A B, construct the polygon P' similar to the given polygon P.

Then P' is the polygon required.

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A B

• § 343 P A' B2 (similar polygons are to each other as the squares on their homologous sides) ;

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Ax. 1

.. 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.

PROPOSITION XXIX. PROBLEM.

362. To construct a polygon similar to a given polygon, and having two and a half times its area.

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À BOM

O -------
Let P be the given polygon.
It is required to construct a polygon similar to P, and
equivalent to 21 P.

Let A B be a side of the given polygon P.
Then VĪ : V2} :: A B : x,
V2 : V5 :: A B : x,

§ 345 (the homologous sides of similar polygons are to each other as the square roots

of their areas). Take any convenient unit of length, as MC, and apply it six times to the indefinite line M N.

On Mo (= 3 M C) describe a semi-circumference ; and on MN (= 6 MC) describe a semi-circumference.

At C erect a I to MN, intersecting the semi-circumferences at D and H.

Then C D is the VĒ, and C H is the V5. § 360 Draw C Y, making any convenient { with C H.

On C Y take C E = A B.

From D draw D E,
and from H draw HY || to D E.

Then C Y will equal x, and be a side of the polygon required, homologous to A B. For CD: CH ::CE : CY,

$ 275 (a line drawn through two sides of a A, II. to the third side, divides the two

sides proportionally).
Substitute their equivalents for C D, C H, and C E;
then

V2 : V5 :: AB :CY. On C Y, homologous to A B, construct a polygon similar to the given polygon P; and this is the polygon required.

Q. E. F.

Ex. 1. The perpendicular distance between two parallels is 30, and a line is drawn across them at an angle of 45°; what is its length between the parallels ?

2. Given an equilateral triangle each of whose sides is 20 ; find the altitude of the triangle, and its area.

3. Given the angle A of a triangle equal to s of a right angle; the angle B equal to į of a right angle, and the side a, opposite the angle A, equal to 10; construct the triangle.

4. The two segments of a chord intersected by another chord are 6 and 5, and one segment of the other chord is 3; what is the other segment of the latter chord ?

5. If a circle be inscribed in a right triangle : show that the difference between the sum of the two sides containing the right angle and the hypotenuse is equal to the diameter of the circle.

6. Construct a parallelogram the area and perimeter of which shall be respectively equal to the area and perimeter of a given triangle.

7. Given the difference between the diagonal and side of a square ; construct the square.

BOOK V.

REGULAR POLYGONS AND CIRCLES.

363. DEF. A Regular Polygon equilateral and equiangular.

is a polygon which is

PROPOSITION I. THEOREM. 364. Every equilateral polygon inscribed in a circle is a regular polygon.

Let A BC, etc., be an equilateral polygon inscribed

in a circle.

We are to prove the polygon A B C, etc., regular.

The arcs A B, BC, C D, etc., are equal, $ 182
(in the same O, equal chords subtend equal arcs).
... arcs A B C, BC D, etc., are equal, Ax. 6
.:. the A, B, C, etc., are equal,

(being inscribed in equal segments). .:. the polygon A B C, etc., is a regular polygon, being equilateral and equiangular.

Q. E. D.

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