Find

Ex. 883. The base and altitude of a triangle are 12 and 20 respectively. At a distance of from the base, a parallel is drawn to the base. the areas of the two parts of the triangle.

Ex. 884. Find the area of a rectangle having one side equal to 6 and a diagonal equal to 10.

Ex. 885. Find the area of a polygon whose perimeter equals 20 ft., circumscribed about a circle whose radius is 3 ft.

Ex. 886. Find the side of an equilateral triangle equivalent to a parallelogram, whose base and altitude are 10 and 15 respectively.

Ex. 887. The sides of two equilateral triangles are 13 and 12 respectively. Find the side of an equilateral triangle equivalent to their difference.

Ex. 888. Two similar polygons have two homologous sides equal to 7 and 24 respectively. Find the homologous side of a third polygon, similar to the given polygons and equivalent to their sum.

Ex. 889. The sides of a triangle are as 8:15:17. Find the sides if the area is 480 sq. ft.

Ex. 890. The sides of a triangle are 8, 15, and 17. Find the radius of the inscribed circle.

Ex. 891. The sides of a triangle are 6, 7, and 8 ft. Find the areas of the two parts into which the triangle is divided by the bisector of the angle included by 6 and 7.

Ex. 892. Find the area of an equilateral triangle whose altitude is equal to h.

PROBLEMS OF CONSTRUCTION

Ex. 893. To construct a triangle equivalent to the sum of two given triangles.

Ex. 894. To transform a rectangle into another one, having given one side.

Ex. 895. To construct a triangle equivalent to the difference of two given parallelograms.

Ex. 896. To transform a square into an isosceles triangle, having a given base.

Ex. 897. To transform a rectangle into a parallelogram, having a given diagonal.

Ex. 898. To divide a triangle into three equivalent parts by lines drawn through a point in one of the sides.

Ex. 899. To bisect a parallelogram by a line perpendicular to a side. Ex. 900. To bisect a parallelogram by a line perpendicular to a given line.

Ex. 901. To divide a parallelogram into three equivalent parts by lines drawn through a vertex.

Ex. 902. To bisect a trapezoid by a line drawn through a vertex.

* Ex. 903. _Divide a triangle into three equivalent parts by lines drawn from a point P within the triangle.

Ex. 904. Divide a pentagon into four equal parts by lines drawn through one of its vertices.

Ex. 905. Divide a quadrilateral into four equal parts by lines drawn from a point in one of its sides.

Ex. 906. Find a point within a triangle such that the lines joining the point to the vertices shall divide the triangle into three equivalent parts. Ex. 907. Construct a square that shall be to a given triangle as 5 is to 4. Ex. 908. Construct a square that shall be to a given triangle as m is to n, when m and n are two given lines.

Ex. 909. Construct an equilateral triangle, that shall be to a given rectangle as 4 is to 5.

Ex. 910. Find a point within a triangle such that the lines joining the point with the vertices shall form three triangles, having the ratio 3:4: 5. Ex. 911. Divide a given line into two segments such that one segment is to the line as √2 is to √5.

Ex. 912. To transform a triangle into a right isosceles triangle.

Ex. 913. Construct a triangle similar to a given triangle and equivalent to another given triangle.

* Ex. 914. Bisect a trapezoid by a line parallel to the bases.

BOOK V

REGULAR POLYGONS. MEASUREMENT OF THE

CIRCLE.

REGULAR POLYGONS

PROPOSITION I. THEOREM

382'. DEF. A regular polygon is one which is equiangular and equilateral.

383. A circle can be circumscribed about any regular polygon.

B

D

Hyp. ABCDE is a regular polygon.

To prove a circle can be circumscribed about ABCDE.

Proof. Construct a circumference through A, B, and C, and

..the circumference passes through D.

In like manner, it may be proven that the circumference passes through the remaining vertices of the polygon.

.. a circle can be circumscribed about the given polygon. Q.E.D.

PROPOSITION II. THEOREM

384. A circle can be inscribed in any regular polygon. HINT. Circumscribe a circle about the given polygon and prove that the center is equidistant from the sides.

385. DEF. The center of a regular polygon is the common center of the circumscribed and inscribed circles of the polygon.

386. DEF. The radius of a regular polygon is the radius of the circumscribed circle.

387. DEF. The central angle is the angle between two radii drawn to the ends of a side.

388. DEF. The apothem of the polygon is the radius of the inscribed circle.

389. COR. 1. The angle at the center of a regular polygon

390. COR. 2. The angle formed by an apothem to a side of a regular polygon, and a radius to an extremity of that side, is

Ex. 915. The lines joining opposite vertices of a regular hexagon pass through the center.

Ex. 916. A triangle is regular if the centers of the circumscribed and inscribed circles coincide.

Ex. 917. A polygon is regular if the centers of the circumscribed and inscribed circles coincide.

Ex. 918. The angle at the center of a regular polygon is the supplement of an angle of the polygon.

PROPOSITION III. THEOREM

391. If the circumference of a circle is divided into any number of equal parts:

(1) The chords joining the points of division successively form a regular inscribed polygon.

(2) Tangents drawn at the points of division form a regular circumscribed polygon.

Hyp. The circumference ACE is divided into the equal arcs AB, BC, CD, etc.

(1) To prove ABCDE is a regular polygon.

(2) To prove tangents drawn at A, B, C, etc., form the regular circumscribed polygon FGHIK.

Proof. GAB = 2 GBA = 2 CBH = HCB, etc., (Why ?)

and

AB=BC CD, etc.

=

(Why?)