Pal'allel Lines and Netwol'ks
2. Write the equation of the line through the origin and the point (-2, 3)
3. Rewrite the equation = so that it will be satis-x a
fied by the coordinates of the origin (0, 0) and at the same time avoid having a fraction with a zero denominator.
SYMMETRY. Two points, P and P', are said to be SYM-METRIC WITH RESPECT TO A LINE if the line is the perpen-dicular bisector of PP'. The line is called the AXIS OF SYMMETRY.
B A
Fig. 24 B
A Notice what happens when
we fold the geometric figure shown in Fig. 24 along the line AB. One-half of the figure then coincides with the other half. We say that this figure is symmetric with respect to the axis AB.
A geometric figure is sym-metric with respect to an axis ifevery point P of the figure (except points on theaxis) has a corresponding point P' in the figure such that PP' is bisected perpendicularly by the axis.
An equilateral triangle has three axes of symmetry.
How many has a square? A rectangle? A regular penta-gon? A regular hexagon?
Show by folding a figure that if two lines intersect, the axes of symmetry of the figure bisect the angles be-tween the lines.
Anequilateral triangle is divided by an axis of symmetry
into two right triangles each of which is said to be sym-metric to the other.
In three dimensions we can have a plane of
analogous to the axis of symmetry in two dimensions.
How many planes of symmetryhas a brick? A cube?
A man? Is there any feature in which two
triangles on a sphere differ from two triangles on a plane?
Figure 25 has noaxisof If, how-ever, we rotate this figure in the plane of the
paper about the point 0, it coincides from time 0 to time with its original position. We say
that this figure is SYMMETRIC WITH RESPECT Fig. 25
TO THE point O. The point 0 is called the
CENTER OF SYMMETRYof the figure.
Has an equilateral triangle a center of symmetry? Has a square? A rectangle? A regular pentagon? A regular hexagon? A brick? A cube?
Bring to class pictures of buildings, church windows, formal designs, and window tracery which show various
kinds of Look for examples of in
nature. Look in a dictionary, an encyclopedia, or a book on trees for pictures of mulberry and sassafras leaves. If you can find pictures of twigs with several leaves on them, notice in what ways they illustrate and in what ways they illustrate lack of Copy the pictures to show to the rest of the class.
Using as a pattern the definition of symmetry with re-spectto an axis given on page 124, state the definition of symmetry with respect to a point in such a way that it will apply to geometric figures with complicated boun-daries, such as the block letter S shown in Fig. 26 on page 126.
Three or more points are said to beCOLLINEAR iftheyall lie on the same straight line. We sometimes say that such 125
points have a line in common. Three or more lines are said to be CONCURRENT if they all pass throughthesame point. We sometimes say that such lines have a point in common.
1. Designers of ornaments make frequent use of axial symmetry, central symmetry, and the close relation of a figure to a network. How many axes of symmetry has each of the seven figures below? Which ones have sym-metry with respect to a point? Which ones show close relation to a network?
Fig. 26
Fig. 27
2. The leaf pictured in Fig. 27 is from poison-ivy. What is there about the shape of poison-ivy leaves that would help you identify this plant outdoors?
*3. Prove that if three or more lines are concurrent, they cut pro-portional segments on two parallel lines.
SeeFig. 28on the next page.
p
s c' s' c'
*4. Prove that ifthree lines cut off proportional seg-ments on two parallel lines, they are either parallel or concurrent. Suggestion: In Fig. 29 assume thatAA'and BB'meet atP and thatBB'and CC'meet atQ. Inorder to prove that P and Q coincide, you must do more than
PB QB
prove that PB'
=
QB'· You must go further and provePB QB . .
that BB'=BB'· In this connection look again at Ex-ercises 20-38, pages64-66.
*5. Prove that if two triangles have their sides re-spectively parallel, they are similar.
*6. Prove thatiftwosimilar triangles have their sides respectively parallel (Fig. 30), the lines AA', BB', CC' joining corresponding vertices are concurrent. Thispoint of concurrence is sometimes called the "center of simili-tude" of two triangles.
127
7. There is one exception to the proposition in Ex. 6 on page 127. What is it? What about AA' and BB' in this case? AA'and CC'?
8. Prove the proposition in Ex. 6 when the point of concurrence lies between the triangles. See Fig. 3I.
e
9. Two lines Iandm (Fig.:
32)intersect at an inaccessible P/i ,/ !
point. It is desired to draw( , I
: : through a given pointP a line
\ I I
m whichifprolonged would pass Fig. 32 through the intersection of I and m. Show how this can be done, assuming that you have a device for drawing a line parallel to a given line.
10. While walking north on Main Street, Fred Harris spies John Reed driving south in a car at 15 miles an hour, just before an elm tree near the curb hides the car from his view. Fred does not want John to see him. How fast must Fred continue walking north in order to keep hidden by the elm tree, if the tree is four times as far from John as from Fred?
11. Prove that the perpendicular bisectors of the sides of a rectangle meet at the point of intersection of the diagonals.
12. Three planes ordinarily intersect in a point, as, for example, the floor and two adjacent walls of room. Under other conditions three planes may have a line in common.
Explain the conditions under which three planes can intersect (two at a time) in three lines; in two lines. When will the planes have no point in common?
In Exercises 13-20 the word is used instead of because in these exercises you are expected prin-cipally to see the relations between the points, lines,
and planes involved and are not required to give the care-ful sort of explanation and justification ordinarily ex-pected of you.
13. Show thatiftwo lines are parallel, every plane con-taining one of the lines, and only one, is parallel to the other. Suggestion: Use the indirect method.
14. Show that two planes perpendicular to the same line must be parallel. Suggestion: Suppose they have a point in common.
15. Show that iftwo parallel planes are cut by a third plane, the lines of intersection are parallel. Suggestion:
Use the indirect method.
16. Show that if two lines are cut by three parallel planes, their corresponding segments are proportional.
SeeFig. 33.
o
Fig. 33 Fig. 34
17. Show that ifa pyramid is cut by a plane parallel to the base, the lateral edges and the altitude are divided proportionally, and the section is a polygon similar to the base. See Fig. 34.
18. Show that ifeach of two intersecting lines is par-allel to a given plane, the plane of these lines is parallel to the given plane. Suggestion: Use the indirect method.
Two lines that do not meet and are not in the same plane are called SKEW LINES.
19. Show that through either of two skew lines it is to pass one plane, and only one, whichis parallel to the other line. Suggestion: Through any point of one of the given lines draw a line parallel to the other given line.
20. Show that through a given point in space one plane, and only one, can be passed parallel to each of two skew lines, or else parallel to one line and containing the other.
21. See ifyou can convince yourself that between any two given skew lines there is one common perpendicular, and only one. Suggestion: Consider:first a random plane parallel to the two given skew lines. Then consider the relation of this plane to all lines that are perpendicular to either of the two skew lines.
SUMMARY
13. Through a given point not on a given line there is one and only one line which does not meet the given line.
DEFINITION: parallel lines
13a. Ifa line meets one of two parallel lines, it meets the other also.
13b. Iftwo lines are parallel to a third line, they are parallel to each other.
DEFINITIONS: system of parallels, transversal
14. Ifa transversal meets two or more lines at the same angle, the lines are parallel.
14a. Lines perpendicular to the same line are parallel.
15. A transversal meets each line of a system of parallels at the same angle.
DEFINITIONS: supplementary angles, complementary angles 15a. Ifa line is perpendicular to one line of a system
of parallels, itisperpendicular to every line of the system.
16. Two transversals are cut into proportional segments by a system of parallels.
DEFINITIONS: parallelogram, trapezoid
17. Perpendiculars to two perpendicular lines are them-selves perpendicular.
DEFINITION: rectangle
17a. The opposite sides of a rectangle are parallel and equal.
DEFINITIONS: rhombus, square; rectangular network, axes, origin, coordinates, slope of a line
18. Iftwo lines have a point in common and have equal slopes, they must coincide.
DEFINITIONS: equation of a line; symmetry with respectto a line, axis of symmetry, symmetry with respect to a point, center of symmetry; collinear points, concurrent lines
The buildingshown above Abbey, the most famous churchin Much of of this building is derived geometric
The insert on tangent which on the of
the
The picture below shows a attempting to scoreby a The companying thal Lhe angle between the goal is Lhe same whether kickeris the side of the field on the 2().yard or
the of the on the line. diagram illustrates
the that all same are equal.
CHAPTER 5