PROBLEMS AND APPLICATIONS.

1. A billiard ball is placed at a point P on a billiard table. In what direction must it be shot in order to return to the same point after hitting all four sides?

(The angle at which the ball is reflected from a

side is equal to the angle at which it meets the side, that is, ≤1 = <≤2, and ≤ 3 = ≤ 4.)

SUGGESTION. (a) Show that the opposite sides of the quadrilateral along which the ball travels are parallel.

(b) If the ball is started parallel to a diagonal of the table, show that it will return to the starting point.

2. Show that in the preceding problem the length of the path traveled by the ball is equal to the sum of the diagonals of the table. 3. Find the direction in which a billiard ball must be shot from a given point on the table so as to strike another ball at a given point after first striking one side of the table.

SUGGESTION. Construct BE to that side of the table which the ball is to strike and make ED = BE.

Cue ball

D

B

B'

D'

H

4. The same as the preceding problem except that the cue ball is to strike two sides of the table before striking the other ball.

5. Solve Ex. 4, if the cue ball is to strike three sides before striking the other ball, also if it is to strike all four sides.

6. In the figure ABCDEF is a regular hexagon. Prove that: (a) AD, BE, and CF meet in a point. (b) ABCO is a rhombus.

(c) The inner circle with center at O and the arcs with centers at A, B, C, etc., have equal radii.

(d) The straight line connecting A and C is tangent to the inner circle and to the arc with center at B.

(e) The centers of two of the small circles lie on the line connecting A and C.

F

E

(f) Find by construction the centers of the small circles.

B

=

D

G

W

T

7. In the figure EG and FH are diameters of the square ABCD. On the diagonals, points K, U, V, W are laid off so that AK: BU CV = DW. EN and SF are in the same straight line, and so on around the figure. Prove that:

Also LM = NP = RS = etc.

(a) KUVW is a square.

(b) AKME and ENUB are equal trapezoids.

H

(c) L, O, 7 lie in a straight line. (d) The four heavy six-sided figures A are congruent.

(g) If AK

n

=

АО

what is the length of ML if the four heavy figures

9 6

occupy half the square ABCD?

8. Find a side of a regular octagon of radius r. See Ex. 3, p. 146.

9. Find the area of a regular octagon of radius r.

ure the lines CH and AD are drawn meeting at K. Prove that ABCK is a parallelogram and find its area if the radius of the circle is r.

12. In the parquet floor design given with Ex. 10, the darker parts are parallelograms constructed as under Ex. 11. What part of the area is of white wood?

13. In the figure ABCD and A'B'C'D' are equal squares, placed as shown. Lines are drawn through A', B', C', and D' parallel to AB, BC, CD, and DA, forming a quadrilateral EFGH.

14. The design opposite consists of white figures constructed like the inner figure preceding, together with the remaining black figures. What part of the figure is white?

15. Show that the altitude of an equilateral triangle with side s is V3.

16. If the angles of a triangle are 30°, 60°, 90°, and if the side opposite the 60° angle is r, show that

60° 90

17. Three equal circles of radius 2 are inscribed in a circle as shown in the figure. Find the radius of the large circle.

SUGGESTION. The center O of the large circle is at the intersection of the altitudes of the equilateral triangle O'0"O"". (Why?)

Hence O'OD is a triangle with angles 30°, 60°, 90° as in Ex. 16.

Solve this problem for

small circles. Ans. Rr +

any radius r of the

2r v3.

3

18. A', B', C', are the middle points of the sides of the equilateral triangle ABC. The sides of the triangle A'B'C' are trisected and segments drawn as shown in the figure.

(a) Prove A"B" C" an equilateral triangle.

(b) Find the area of A"B"C" and of the dotted hexagon.

(c) Find the area of the triangle GFC and of the trapezoid DEB"A".

19. In the figure ABC is an equilateral triangle. On the equal bases PM, MD, DE equilateral arches are constructed, two of them tangent to the sides of the triangle at L and N respectively. Circles O' and O" are each tangent to a side of the triangle and to two of the arches. Circle O is tangent to circles O' and O" and to both sides of the triangle.

(a) If AB = a, find DE.

SUGGESTION. In the right triangle DBL one acute angle is 60°.

(b) Find the ratio r: r', r being the base DE of the arch and r' the radius of the circle O'.

(c) By what fraction of a will the circles O' and O" fail to touch each other?

(d) Find the radius of the circle 0.

20. In the figure CD = DA = AG = GB, and DE EA AF= FG. Semicircles are constructed on the diameters CB, CG, CD, DG, DB, GB. HG = KD = CE. Arcs GL and DM have centers H and K respectively.

Circles are constructed tangent to the various arcs as shown in the figure. Thus O O" is tangent to semicircles on the diameters CB, CG, and DB. O' is tangent to the semicircles on the diameters CG and

DB and to the arc DM. Let CB: = a.

(a) Find the areas of each of the six semicircles.

(b) Find the radius r''.

SUGGESTION. EA, EP and AN are known.

(c) Find the radius r'.

SUGGESTION. Enumerate the known parts in A EDO' and FDO'. (d) Find r and r1v.

(e) Having determined the radii of the various circles, show how to construct the whole figure.

(f) What fraction of the area of the whole figure is occupied by the six circles?

21. Prove that two segments drawn from vertices of a triangle to points on the opposite sides cannot bisect each other.