12. Construct a triangle similar to a given triangle and having 16 times the area.

13. The middle points of the sides of a quadrilateral are connected. Show that the area of the parallelogram so formed is half the area of the quadrilateral.

14. ABCD is a square. Each side is divided into four equal parts and the construction completed as shown in the figure.

(a) Prove that QNOH, KLOE, etc., are parallelograms.

(b) What part of the area of the square is occupied by these four parallelograms?

(c) What part of the area of the square is occupied by the four triangles NEO, LGO, PFO, and MHO?

(d) If AB = 6, find the lengths of KO and QP.

(e) Find the ratio of the segments KO and QP. Does this ratio depend upon the length of AB?

D

M

H

F

E

A

K B

Parquet Flooring.

(f) In the parquet floor design what fraction is made of the dark wood? Does this depend upon the size of the original square?

15. On a given line-segment AB as a hypotenuse construct a right triangle such that the altitude upon the hypotenuse shall meet it at a given point D.

16. If ABC is a right triangle and CD LAB,

prove that

AC
CB2

=

AD
ᎠᏴ

B

17. By means of Exs. 15 and 16 construct two segments HK and LM such that the ratio of the squares on these segments shall equal a given ratio.

18. Divide a given segment into two segments such that the areas of the squares constructed upon them shall be in a given ratio.

19. On a given segment AB find a point D such that

AB

=

= 2.

AD2

20. Construct a line parallel to the base of a triangle such that the

resulting triangle and trapezoid shall be equivalent.

21. Construct two lines parallel to the base of a triangle so that the resulting two trapezoids and the triangle shall have equal areas.

....

A

-B

C

-D

Points E, F, G, are so taken that

...

23. ABCD is a square. AE: = AF GB = BH = = (a) If AB = 6 and AF = 1, find the sum of the areas of the four triangles EFO, GHO, KLO, MNO.

(b) Find the sum of these areas if AB = a and AF=h.

(c) If AB = 6 and if the sum of the areas of the triangles is 9 square inches, find AF.

a2

find

n

(d) If AB = a and if the sum of the areas of the triangles is AF. Interpret the two results.

24. Show how to construct a square whose area is n times the area of a given square.

25. Construct a triangle similar to a given triangle and equivalent to n times its area.

26. Construct a hexagon similar to a given hexagon and equivalent to n times its area.

27. Show how to construct a polygon similar to a given polygon and equivalent to n times its area.

28. The alternate middle points of the sides of a regular hexagon are joined as shown in the figure.

(a) Are the triangles thus formed equilateral? Prove.

(b) Is the star regular (i.e. are its six acute angles equal and its sides equal)?

(c) Compare the three segments into which each triangle divides the sides of the other.

(d) Is the inner hexagon regular? Prove. See Ex. 1, p. 76.

(e) If AB:

= 6 in., find the area of the large hexagon, the star, and the small hexagon.

29. A border is to be constructed about a given square with an area equal to one half that of the square.

(a) By geometrical construction find the outer side of the border if the side of the square is given.

(b) If an outer side of the border is 24 in., find a side of the square, its area being two thirds that of the border.

30. A border is constructed about a given regular octagon, such that its area is equal to that of the octagon.

(a) If a side of the given octagon is a given segment AB, find by geometrical construction a segment equal to an outer side of the border.

(b) If a side of the given octagon is 16 in., find an outer side of the border.

Ceiling Pattern.

Ceiling Pattern.

31. The accompanying design is based on a set of squares such as ABCD. The small triangles are equal isosceles right triangles constructed as shown.

(a) Are the vertical and the horizontal sides of the octagons equal?

(b) Are the octagons regular?

(c) If a side of one of the squares is 6, find the area of one of the octagons.

(d) What fraction of the whole tile design is occupied by the light-colored tiles? Does this depend upon the size of the original squares?

32. Given two lines at right angles to each other. Find the locus of all points such that

D

Α

B

the sum of the squares of the distances from the lines is 25.

33. Given two concentric circles whose radii are r and r'. Find the length of a chord of the greater which is tangent to the smaller.

34. If two equal circles of radius r intersect so that each passes through the center of the other, find the length of the common chord.

35. The square on the hypotenuse of a right triangle is four times the square on the altitude upon the hypotenuse. Prove it isosceles.

36. In a right triangle the hypotenuse is 10 feet and the difference between the other sides is 2 feet. Find the sides.

37. Two equal circles are tangent to each other and each circle is tangent to one of two lines perpendicular to each other. Find the locus of the points of tangency of the two circles.

SUGGESTION. Note that the point of tangency bisects their line of centers and that the centers move along lines at right angles to each other.

38. The square on a diagonal of a rectangle is equal to half the sum of the squares on the diagonals of the squares constructed on two adjacent sides of the rectangle.

39. Show that the diagonals of a trapezoid form with the nonparallel sides two triangles having equal areas.

CHAPTER V.

REGULAR POLYGONS AND CIRCLES.

REGULAR POLYGONS.

332. A regular polygon is one which is both equilateral and equiangular.

According to this definition, determine whether each of the following polygons is regular or not and state why:

An equilateral triangle, an equiangular triangle, a rectangle, a square, a rhombus. Draw a figure to illustrate each. Make a triangle which fulfills neither condition of the definition, also a quadrilateral.

333. The general problem of constructing a regular polygon depends upon the division of a circle into as many equal parts as the polygon has sides.

The problem of dividing the circle into equal parts can be solved in some cases by the methods of elementary geometry, and some of these methods will be considered in this chapter. In most cases this problem cannot be solved by elementary methods.

E.g. the circle may be divided into 2, 3, 4, 5, 6, 8, 10, 12, 15, equal parts, but not into 7, 9, 11, 13, 14, equal parts.

If a circle has already been divided into a certain number of equal parts, it may then be divided into twice, four times, eight times, etc., that number of parts by repeated bisection of the arcs. (See § 204, Ex. 1.)

The division of the circle into equal parts depends upon the theorem (§199) that equal central angles intercept