Prove that the lines joining the mid-points of the

sides of a quadrilateral form a parallelogram whether or not the four vertices of the quadrilateral are in the same plane.

c

P D 29. Perpendiculars are drawn to a random line from the four vertices of a parallelogram. Express the length of the perpendicular drawn to this same line from the point of intersection of the diagonals in terms of the lengths of the perpendiculars from the vertices.

Theorem 17. Perpendiculars to two perpendicular lines are themselves perpendicular.

GIVEN: OA perpendicular to OB, AC perpendicular to OA, and BD perpendicular to OB.

See Fig. 17.

TO PROVE: ACperpendicular to BD.

PROOF: We must first show 0 A

that ACand BDhave a point Fig. 17 in common. AC is parallel to

OB, because two lines perpendicular to the same line are parallel. BD, since it meets OB at B, must meet ACat some point P, for otherwise we should have two lines through B parallel to AC.

The transversal BD and the parallel lines OB and AC form the equal anglesOBD and APD. ConsequentlyAC and BD are perpendicular. Why?

Each angle of the quadrilateral AOBP is a right angle.

Such a quadrilateral is called aRECTANGLE.

Prove the following corollary.

Corollary 17a. The opposite sides of a rectangle are parallel and equal.

A RECTANGLE is an equiangular .parollelogram; an equi-lateral parallelogram is called a RHOMBUS. A parallelo-gram which is both and equilateral is called a

SQUARE.

RECTANGULAR NETWORK. We have seen that all the lines perpendicular to a given line form a system of par-allels. See Fig. 18. The lines perpendicular to two per-pendicular lines will form two systems of parallels; by Theorem 17 every member of one system will be perpen-dicular to every member of the other.

v

Fig. 18

.5 : 2

C

Fig. 19

x

The collection of lines perpendicular two perpendicular lines is called a RECTANGULAR NETWORK (Fig. 19). The two given perpendicular lines are called the AXES of the net-work; their intersection 0 is called the ORIGIN. Every line

perpendicular to either one of the axes belongs to the network. Every point P in the plane lies on one and only one line of each system of parallels in a rectangular net-work.

Any pair of perpendicular lines each of which is per-pendicular to an axis of the network may serve as axes of the network, and then their intersection becomes the origin. The network erected on the new axes will be identical with the original network. The choice of the particular pair of perpendicular lines which shall serve as axes of the network is determined for the most part by convenience and by the particular task in hand; this choice of axes can be changed, ifdesired, during the course of a demonstration.

No line not of the network can be parallel to any line of the network; for ifparallel it would then be a line of the network.

COORDINATES. The distances PD and PC measured along lines of the network (Fig. 19 on page 118) are the dis-tances from point P to the axes; they are equal to OC and OD respectively. The directed distances OC and OD locate the point P in relation to the two axes; they are called theCOORDINATESofP.

IfOC=5 and OD=3, the coordinates ofP are the num-bers 5 and 3. The coordinates of a point are often writ-ten after the letter designating the point, like this:

P:(5, 3). In Fig. 19 notice that the coordinates of P and of P' are shown in this way.

The axis OX is often called the x-axis, and the axis OY is often called the y-axis, Hence the coordinate of P which is measured along the x-axis is called the x-coordi-nate of P; and the which is measured along they-axis is called the y-coordinate of P. In Fig. 19 the x-coordinate of P is 5, and the y-coordinate is 3. What

is the z-coordinate of P'? Whatisthe y-coordinate ofP'?

Notice that the z-coordinate is always written before the y-coordinate, For every point in a plane there are two numbers or which locate the point on a given network.

THE SLOPE OF A LINE. In Fig. 20 the of the point P are a and b. Notice that the relative size of a and b determines the shape of the rectangle CODP and the inclination of the line OP to the axes. The ratio

a is called the SLOPE of the line OP with respect to the x-axis. If a=5 and b =3, then the slope of OP is or 0.6. could just as well say that the slope of OP with respect to the y-axis is a but ordinarily we shall think of slopes with respect to the x-axis only.

Ifb =0, the slope of OP is 0. Ifa=0, we cannot speak of the slope of OP with respect to the x-axis because we cannot divide by zero.

y

Every line parallel toOP, such as QR in Fig. 20, forms an angle of the same size with the x-axis as OP forms.

This means that the right triangles QRE and OPC are

similar and that the slope of QR can be written either as ERQE'ta mgki Qas ongm, or as. . b We see, there ore,f th ta every line parallel to OPhas the same slope as OP.

EXERCISES

1. Locate the following points on a rectangular net-work, taking one-quarter of an inch or any other con-venient distance as the unit of measurement: A:(3, 5);

B:(4, 2); C:(2, 3); D:(6.50, 2.75).

2. Draw a line through each of these points and the origin and find its slope.

3. Locate each of the following points on your network:

E:(-5, 3); F:(5, -3); G:(-5, -3); H:(-4, 2);

1:(2, -5); J:(-2, -3.5); K:(-3.5, -2.5).

4. Find the distance from the origin to each of the points mentioned in Exercises 1 and 3. (Use the Pythag-orean Theorem.)

5. Compute the following distances: AC, BD, AG, EF, EK, CJ, JK, ID, KD, JD. (Use the Pythagorean Theorem.)

6. Draw a line through the origin and the point Hand find its slope. Notice that one of the coordinates of H is negative.

7. Draw the lines CA, EH, and HFand find the slope of each of these three lines.

8. Compare the slopes of lines GK and BD. What is the slope of line HB?

9. Pick out the line segment on your network that has the steepest slope. Pick out a line segment on your network that has no slope.

y

Theorem 18. If two lines have a point in common and have equal slopes, they must coincide.

GIVEN: LinesPQand PRhaving equal slopes and pass-ing through P. SeeFig. 21.

TO PROVE: PQand PR must coincide.

y' R

ANALYSIS: To prove that these lines coincide, we must show that they make the same angle with some line of the network. We can prove angles equal by means of the three Cases of Similarity; the equal slopes suggest sides of right triangles in proportion and the advisability of trying to apply Principle 5.

PROOF: If point P is not at the origin of the network to which the slopes are referred, we have only to choose P as a new origin and choose for new axes the lines of the network which pass through P. Complete the proof.

EQUATION OF ALINE. The slope of the line through the origin 0 and the point P is given in terms of the of P. In Fig. 22 on the next page the slope of OP But the slope of this line can be stated

a

equally well in terms of the (x, y) of any other

(a,b)

point on the line, such as the point V. That is, the slope of DV is equal to the slope of OP; for triangles LOP and MOV are similar, by Case 2 of Similarity (Principle 6). It follows that This

equa-x a

tion is called THE EQUATION OF THE LINE OP. Any point

except 0 on OP must have coordinates (x, y) which satisfy this equation, and by Theorem 18 any point whose coordinates (x, y) satisfy this equation must lie on OP.

Thus the equation = is an algebraic way of saying that x a

the ratios of the coordinates of all points except (0, 0) on lineOP are equal.

Ifthe point 0 does not happen to be at the origin, it is always possible to choose a new pair of axes for the net-work so that the origin shall fall at O.

The equations of lines that are members of the network can be obtained directly with-out using slopes. For exam-ple, in Fig. 23 notice the line of the network that is parallel to the x-axis and passes through the point (a, b). The y-coordinate of every point

on this line is b; therefore the equation of this line is y=b. Similarly, since the x-coordinate of every point on the line through (a, b) perpendicular to the x-axis is a, the equation of this line is x

=

1. Write the equation of the line through the origin and the point (5, 4).

2. Write the equation of the line through the origin