Ex. 609. Construct an isosceles triangle on a given base and equivalent to a given trapezium.

Ex. 610. Construct a right triangle equivalent to a given triangle, having given one of the sides about the right angle.

Ex. 611. Construct a right triangle equivalent to a given triangle, having given the hypotenuse.

Ex. 612. Construct a triangle equivalent to a given triangle and having its base and altitude equal.

Ex. 613. Construct an equilateral triangle equivalent to a given triangle. Ex. 614. Construct an equilateral triangle equivalent to a given square. Ex. 615. Construct a rectangle having a given diagonal and equivalent to a given rectangle.

Ex. 616. Construct a rectangle having a given diagonal and equivalent to a given square.

Ex. 617. Construct a square equivalent to a given trapezoid.

Ex. 618. Construct a triangle equivalent to a given trapezoid.

Ex. 619. Construct a parallelogram equivalent to a given trapezoid and having for its base the longer base of the trapezoid.

Ex. 620. Construct a triangle equivalent to a given triangle and similar to another given triangle.

Ex. 621. Construct a parallelogram equivalent to the sum of two given parallelograms.

Ex. 622. The area of a square is 16. Construct a square that shall be to

it in the ratio of 5 to 3.

Ex. 623. Construct a hexagon similar to a given hexagon, having its ratio to the given hexagon as 5 is to 3.

Ex. 624. Construct a square equivalent to two thirds of a given hexagon. Ex. 625. Construct a square equivalent to the sum of a given pentagon and a given parallelogram.

Ex. 626. Divide a given triangle into two equivalent parts by a line perpendicular to one side.

Ex. 627. Divide a given triangle into two equivalent parts by a line parallel to one side.

Ex. 628. Bisect a given trapezoid by a line parallel to the bases.

Ex. 629. Bisect a given quadrilateral by a line drawn from one of the vertices.

Ex. 630. Bisect a given quadrilateral by a line drawn from any point in its perimeter.

Ex. 633. Given the sides of a triangle, to compute the

Ex. 634. Given the sides of a triangle, to compute the bisectors of the angles.

Solution. Circumscribe a circle about ▲ ABC; produce CD to meet the circumference in E; and draw BE.

E

Ex. 635. Given the sides of a triangle and the radius of the circumscribing circle, to compute the area of the triangle.

Solution. Denote the radius of the circle by r.

Ex. 636. Given the sides of a triangle, to compute the radius of the circumscribed circle.

b

b

whence,

r =

abc

4√s(s − a) (s - b) (s — c)

Ex. 637. The three sides of a triangle are 58 ft., 51 ft., and 41 ft. in length. What is the area of the triangle?

Ex. 638. Find the altitude on each of the sides of a triangle whose sides are respectively 7 in., 9 in., and 11 in.

Ex. 639. If the sides of a triangle are respectively 4m, 6m, and 8m long, what are its three medians ?

Ex. 640. What is the area of a triangle, if the radius of the circumscribing circle is 6.196m and the sides of the triangle are respectively 9m, 6m, and 12m in length?

Ex. 641. The sides of a triangle are respectively 12 in., 11 in., and 9 in. in length. Find the radius of the circumscribing circle.

Ex. 642. The sides of a triangle are respectively 30dm, 50dm, and 70dm. Find the lengths of the three angle bisectors.

Ex. 643. If two sides and one of the diagonals of a parallelogram are respectively 12 in., 15 in., and 18 in., what is length of the other diagonal? What is the area of the parallelogram?

BOOK VI

REGULAR POLYGONS AND MEASUREMENT OF THE CIRCLE

374. A polygon which is equilateral and equiangular is called a Regular Polygon.

An equilateral triangle and a square are regular polygons.

Proposition I

375. Draw a circle and inscribe in it any equilateral polygon. How do the arcs subtended by the sides of the polygon compare? How do the arcs intercepted by the sides of the angles of the polygon compare? How do the angles themselves compare? What may any equilateral polygon that is inscribed in a circle be called?

Theorem. Any equilateral polygon inscribed in a circle is a regular polygon.

Data: Any equilateral polygon, as E ABCDE, inscribed in a circle.

To prove ABCDE a regular polygon.

D

B

Proof. §-196, arc AB = arc BC= arc CD =

= etc.;