CHAPTER VIII

METRIC RELATIONS

DERIVATION OF FORMULÆ

440. Constant and variable magnitudes. A geometric magnitude is called a constant or a variable, according as its value remains fixed, or changes, during a given discussion.

Thus consider the radius OP of a circle and its projection OQ

on any diameter AB when the point P moves around the circle. It is clear that OP remains fixed in length, while OQ varies from a length equal to the radius when P is at B to zero when OP is perpendicular to AB; then increases again to a length equal to a radius when P is at A, etc. Hence, in this discussion OP is a constant and OQ is a variable.

A

P

D

FIG. 202.

B

441. A function of a variable. If two variables are so related that a change in one produces a corresponding change in the other, each is called a function of the other.

Thus an angle and its tangent, sine, or cosine are so related that a change in the angle produces a corresponding change in each of the ratios mentioned; and conversely. Hence, the sine, cosine, and tangent of an angle are called functions of the angle.

NOTE. If a variable is a function of another variable, it should not be assumed that a certain change in the one produces an equal change in the other. For example, the area of a square is a function of the side (that is, if the side is changed, the area is changed); but a change in the side produces a much greater change in the area, comparatively. Thus, if the side is doubled, the area will be increased four times; if the side is increased 10 times, the area will be increased 100 times, etc.

192

This fact is of great importance in engineering. Doubling the size of a beam, for example, does not necessarily double the strength. Many of the great engineering disasters, such as the collapse of ponderous bridges, buildings, etc., have had their causes traced to mistakes of this nature.

442. A formula. A metric relation expressing a given variable as a function of one or more variables is called a formula.

Thus sah is the formula expressing the area of a rectangle as a function (in terms) of two adjacent sides. In this formula s is said to be expressed explicitly in terms of a and h, and is called an explicit function of a and h; either a or h is called an implicit function of the other two.

To solve an equation for a given variable is to express this variable as an explicit function of the other variables involved.

Before substituting in a formula to find the numerical value of a variable corresponding to particular values of the other variables involved always express the unknown variable explicitly in terms of the other variables where possible.

1. Given s= bh. Find b, when s = 40 and h = 10.

2. Given A =}(a+b)h: (a) Find h, when A = 100, a = 8, and b = (b) find a, when A

120, b = 24, and h

= 6.

12 ;

3. The base b of a triangle is increased by an amount equal to k. By how much must the altitude h be decreased so that the area may remain constant ?

4. If a rectangle varies, keeping one side fixed, how does its area vary with respect to the adjacent side?

5. If a polygon varies in size, keeping the same shape, how does its area vary with respect to its sides?

6. If a man 6 feet high can see 3 miles on a smooth sea, can he see 100 miles from an elevation of 200 feet?

SUGGESTION: See Ex. 3, § 437.

7. The strength of a rectangular beam varies directly as the breadth multiplied by the square of its depth. Compare the strengh of a beam 4 in. in breadth and 3 in. in depth with that of a beam 3 in. in breadth and 4 in. in depth.

THE RIGHT TRIANGLE

444. THEOREM. In a right triangle each leg is the mean proportional between the hypotenuse and its projection on the hypotenuse.

Given

FIG. 203.

the right AABC with the right angle at C, and p and q the projections of a and b on c, respectively.

445. COROLLARY 1.

Why?

Why?

In a right triangle the square of the hy

potenuse is equal to the sum of the squares of the legs.

For, from step 2, a2 = cp, and b2

= cq.

.. a2 + b2 = cp + cq = c (p + q) = c2.

(Compare with § 366.)

Why?

446. COROLLARY 2. In a right triangle the square of either leg is equal to the difference between the square of the hypotenuse and the square of the other leg.

447. COROLLARY 3. The altitude drawn to the hypotenuse of a right triangle divides the given triangle into two triangles which are similar to the given triangle and to each other.

448. COROLLARY 4. The altitude drawn to the hypotenuse of a right triangle is the mean proportional between the projections of the legs on the hypotenuse.

449.

For, since rt. ▲ CDA ~rt. A CDB (Cor. 3),

1. If two chords of a circle are equal, they are equally distant from the center, and conversely; if they are unequal, the greater is the less distance from the center, and conversely.

Compare d and d', if a' = a; if a' >a; if a' <a; and conversely.

2. The longest chord that can be drawn through a given point within a circle is the diameter drawn through that point; and the shortest chord is perpendicular to the diameter through that point.

3. A chord drawn from any point on a circle to either extremity of a diameter is the mean proportional between the diameter and its projection on the diameter.

4. The perpendicular from any point on a circle to a diameter is the mean proportional between the segments into which it divides the diameter. 5. The difference between the squares of two sides of any triangle is equal to the difference between the squares of the projections of these sides on the third side.

SUGGESTION b2 — q2 = h2 = a2 — p2.

Why?

6. The squares of the two legs of a right triangle B are to each other as their projections on the hypote

nuse.

SUGGESTION: From step 2, § 444, a2 = cp, and b2 = cq.

7. The squares of two chords drawn from the same point on a circle have the same ratio as the projections of the chords on the diameter drawn from the same point. A

8. The sum of the squares of the segments of two perpendicular chords is equal to the square of the diameter of the circle.

SUGGESTION: Apply § 445 and prove DF = BC.

450. THEOREM. In a right triangle:

1. Either leg is equal to the product of the hypotenuse into the sine of the opposite angle or the cosine of the adjacent angle.

II. Either leg is equal to the product of the other leg into the tangent of the angle opposite the former leg.

SUGGESTION: See definitions of sine, cosine, and tangent, § 420.

451. COROLLARY 1. The sine of an acute angle is equal to the cosine of its complement.

For

a

sin A == cos B: = cos (90° — A).

с

452. COROLLARY 2. The tangent of an acute angle is the reciprocal of the tangent of its complement.

453. Use of a table of sines, cosines, and tangents. On page 218 is exhibited for reference a three-place table of sines, cosines, and tangents of acute angles at intervals of one degree. The use of the table may be described as consisting of two converse operations as follows: (1) Given an angle expressed in degrees and minutes to obtain its sine, cosine, or tangent from the table; and (2) given the sine, cosine, or tangent expressed as