337.

ORAL AND REVIEW QUESTIONS

How is the center of a regular polygon found? Tell a second way. To what are the perimeters of regular polygons proportional? the areas? the circumferences of circles? the areas of circles? Give a formula for the circumference of a circle, two formulas for the area of a circle. To what is the side of a regular inscribed hexagon equal? the side of a regular inscribed decagon? the side of a circumscribed square? of an inscribed square? of a circumscribed equilateral triangle? of an inscribed equilateral triangle? What must be known to prove an inscribed polygon regular? a circumscribed polygon? Explain the inscribed case. What is the construction upon which the regular decagon depends? How can a regular pentagon be drawn? Tell which of the following regular polygons can be constructed with compass and ruler, and explain; 7, 8, 9, 12, 14, 16 sides. What values of π are most commonly used? What is the circumference of a circle of radius 10? the area? How can you find the radius from the circumference? the radius from the area? the circumference from the area? the area from the circumference? How can circumferences be added? subtracted? multiplied? How can the areas of circles be added? subtracted? multiplied? divided? Explain how to draw a circle having its circumference double that of a given circle; its area double that of the given circle.

GENERAL EXERCISES

509. The area of a regular dodecagon (12) equals three squares on its radius.

510. What is the radius of a circle, if the area of the regular inscribed hexagon is 6√3?

511. The area of the ring between two concentric circles equals that of a circle whose diameter is that chord of the larger which is tangent to the smaller.

512. If two chords of a circle are perpendicular to each other, the sum of the circles on the sects as diameters equals the original circle.

513. On the sides of a square of side a, as diameters, circles are drawn. Find the area of the parts into which the square is divided.

514. On the sides of an equilateral triangle of side a as diameters, circles are drawn. Find the areas of the parts of the figure formed.

515. With the vertices of an equilateral triangle of side a as centers, and a radius equal to half the side, circles are drawn. Find the area of the entire figure, and of the figure inside the triangle bounded by the arcs.

516. Construct a regular octagon on a given sect as side.

517. Construct a regular hexagon on a given side.

518. Construct a regular decagon on a given side.

519. Construct a regular pentagon on a given side.

520. Construct a regular hexagon, given the shorter diagonal, 521. Construct a circle equal to the sum of two given circles. 522. Construct a circle equal to the difference of two given circles. 523. Construct a circle equal to the sum of any number of given circles.

524. Construct a circle equal to any number of times a given circle. 525. Construct a circumference equal to the sum of two given circumferences.

526. Construct a circumference equal to the difference of two given circumferences.

527. Construct a circumference equal to the sum of any number of given circumferences.

528. Construct a circle whose area has any given ratio to the area of a given circle.

529. Construct a circle whose area is one half the area of a given circle.

530. Divide a given circumference into parts having the ratio 3:7, by a line through a given point.

531. Divide the surface of a circle into equal parts by a concentric circle.

532. Divide the surface of a circle into any number of equal parts by concentric circles.

533. Given the radius of a circle, find the perimeters of regular inscribed and circumscribed polygons of 3, 4, 5, 6, 8, 10 sides.

534. Cut off two thirds of a circle by a line through a given point.

GENERAL

THE FORMULAS OF GEOMETRY

338. I. IN AN n-SIDED POLYGON.

(1) The sum of the interior angles equals (n 2) st. 4. (2) The sum of the exterior angles equals 2 st. 4.

II. ANGLES FORMED BY LINES MEETING A CIRCUMFER

ENCE.

(1) Vertex on the circumference, measured by half the arc; includes inscribed angles (three cases), tangent and chord angles.

(2) Vertex inside the circle, measured by half the sum of the arcs.

(3) Vertex outside the circle, measured by half the difference of the arcs; includes angles between two secants, two tangents, tangent and secant; angle between two tangents supplemental to central angle between radii to points of tangency.

III. AREA.

(1) Rectangle, equals base times altitude.

(2) Parallelogram, equals base times altitude.
(3) Triangle, equals half base times altitude..

(4) Triangle, given the sides, equals the square root of
s(s − a) (s — b) (s −c), where s is half the sum of
the sides.

(5) Equilateral triangle, equals the square of half the side times the square root of three.

(6) Trapezoid, equals half the sum of the bases times the altitude.

(7) Regular polygon, equals half the perimeter times the apothem.

(8) Circle, equals π times the square of the radius.

IV. EQUIVALENCE FORMULAS BASED ON THE AXIOM OF THE

WHOLE.

(1) The square on the sum of two sects is equivalent to
the sum of their squares plus twice their rectangle.
(2) The square on the difference of two sects is equivalent
to the sum of their squares less twice their triangle.
(3) The square on twice a sect is equivalent to four
squares on the sect.

(4) the difference of the squares on two sects is equiva-
lent to the rectangle of their sum and their difference.

V. THE SQUARE ON A SIDE OF A TRIANGLE.

(1) Opposite a right angle, is equivalent to the sum of the squares on the other sides.

(2) Opposite an obtuse angle, is equivalent to the sum of the squares on the other sides plus twice the rectangle of one side by the projection of the other on its line. (3) Opposite an acute angle, is equivalent to the sum of the squares on the other sides, less twice the rectangle of one side by the projection of the other side on its line; or, in a right triangle, to difference of squares. VI. THE SUM OF THE SQUARES ON TWO SIDES OF A TRIANGLE. Is equivalent to twice the sum of the squares on one half the third side and on the median to that side; difference, to two rectangles of the base by its median's projection.

VII. PROPORTIONS BETWEEN SECTS.

(1) Parallels cutting transversals, including triangle case. (2) Sides of similar polygons are proportional.

(3) In a right triangle with the altitude to the hypotenuse, (a) the altitude is the mean between the sects of the hypotenuse;

(b) either leg is the mean between the hypotenuse and the projection of that leg on the hypotenuse.

(4) If two secants cut a circle, the product of the sects from the vertex is the same; a tangent is the mean between the sects of a secant from the same point.

(5) The bisector of an angle of a triangle cuts the opposite side (internally or externally) into sects proportional to the sides including the angle.

(6) Perimeters of regular polygons are proportional to sides, apothems, or radii.

(7) Circumferences are proportional to radii.

VIII. PROPORTIONS BETWEEN AREAS.

(1) Areas of triangles having an angle equal are propor-
tional to the product of the sides including the angle;
if the triangles are equivalent, the product of the sides
in one equals the product in the other, and conversely.
(2) Areas of any similar polygons are proportional to the
squares of corresponding sides.

(3) Areas of circles are proportional to the squares of their
radii.

IX. THE CIRCUMFERENCE OF A CIRCLE equals 2 πR.
X. ALTITUDE.

(1) Of any triangle, is found from III, 4, or by first find-
ing the projection in V, (2) or (3).

(2) In an equilateral triangle, equals half the side times the square root of three.

XI. PROJECTION of a Side of a TRIANGLE. from V, (2), or (3). XII. MEDIAN OF A TRIANGLE, from VI.