374. THEOREM. If a line is parallel to one side of a triangle and cuts the other two sides, then it divides these two sides in the same ratio.

Graphic representation:

Lay off CA along the horizontal axis and CB along the vertical axis, thus locating the A point P1.

In like manner find P2 with the coördinates CD and CE. If CD and CA are commensurable, we know that

CD CE

=

CA CB

D

C

Hence 0, P2 and P1 are collinear.

ΤΑ

If CD and CA are incommensur

able, show, as in § 368, that the corresponding point P lies on the line OP, and hence, as in that case, also

375. PROBLEM. To represent graphically the relation between the area and side of a square as the side varies continuously.

SOLUTION. On the horizontal axis lay off the segments equal to various values of the side s, and on the vertical axis lay off segments equal to the corresponding areas A.

(1) If one horizontal space represents one unit of length of side, and one vertical space one unit of area, then the points P1, P2, P3, etc., are found to lie on the steep curve.

(2) If five horizontal spaces are taken for one unit of length of side and one vertical space for one unit of area, then the points P1', P2', P', etc., are found and the less steep curve is the result.

The student should locate many more points between those here shown and see that a smooth curve can be drawn through them all in each case.

The graph of the relation between two variables, one of which varies as the square of the other, is always similar to the one here given.

376. The area of a square is said to vary as the square of one of its sides, that is, A = s2.

For example, the theorem: The areas of two similar polygons are in the same ratio as the squares of any two corresponding sides, means that if a given side of a polygon is made to vary continuously while the polygon remains similar to itself, the area of the polygon varies continuously as the square of the side.

1. From the last graph find approximately the areas of squares whose sides are 3.4; 5.25; 6.35.

2. Find approximately from the graph the side of a square whose area is 28 square units; 21 square units; 41.5 square units.

3. Construct a graph showing the relation between the areas and sides of equilateral triangles.

4. Given a polygon with area A and a side a. Construct a graph showing the relation between the areas and the sides corresponding to a in polygons similar to the one given.

5. From the graph constructed in Ex. 3, find the area of an equilateral triangle whose sides are 4. Also of one whose sides are 6. Compute these areas and compare results.

6. Construct a graph showing the relation between a side and the area of a regular hexagon. By means of it find the area of a regular hexagon whose sides are 6. Compare with the computed area.

378. PROBLEM. To construct a graph showing the relation between the radius and the circumference of a circle, and between the radius and area of a circle, as the radius varies continuously.

SOLUTION. Taking ten horizontal spaces to represent one unit of length of radius, and one vertical space for one unit of circumference in one graph and one unit of area in the other, we find the results as shown in the figure.

1. From the graph find approximately the circumference of a circle whose radius is 2.7, also of one whose radius is 3.4.

2. Find the radius of a circle whose circumference is 17, also of one whose circumference is 23.

3. Find the areas of circles whose radii are 1.9, 2.8, 3.6.

4. Find the radii of circles whose areas are 13.5, 25.5, 37, 45.

5. How does the circumference of a circle vary with respect to the radius?

6. How does the area of a circle vary with respect to the radius? 7. Find the radius of that circle whose area in square units equals its circumference in linear units.

DEPENDENCE OF VARIABLES.

380. In the preceding pages we have considered certain areas or perimeters of polygons as varying through a series of values. For example, if a rectangle has a fixed base and varying altitude, then the area also varies depending on the altitudes. The fixed base is called a constant, while the altitude and area are called variables.

The altitude which we think of as varying at our pleasure is called the independent variable, while the area, being dependent upon the altitude, is called the dependent variable.

381. The dependent variable is sometimes called a function of the independent variable, meaning that the two are connected by a definite relation such that for any definite value of the independent variable, the dependent variable also has a definite value.

Thus, in A

=

s2 (§ 376), A is a function of s, since giving s any definite value also assigns a definite value to A.

Similarly, C is a function of r in C = 2 πr, and A is a function of r in 4 = πr2.