Construction 2

305. Construct a fourth proportional to three given lines.

Given the lines a, b, and c.

Required to construct a line x, so that a:b= c : x.

Construction. Draw AH and AL, making angle A about 30°. On AH lay off AR equal to a, on RH lay off RK equal to c, and on AL lay off AB equal to b. Draw BR, and through K construct KC parallel to BR, cutting AL at C. Then BC is the required fourth proportional.

The proof should be supplied by the student.

EXERCISES

ab

205. Construct a if x is equal to a, b, and c being lines of given length.

с

206. Construct x if x is equal to ab, a and b being lines of given length.

207. Construct a third proportional to a and b, two given lines. 208. Construct x if a2

=

bx, a and b being two given lines. 209. Construct x if a:b=x: c where a, b, and c are lines of given length.

Construction 3

306. Divide a given line into parts proportional to two or more given lines.

Given the lines a, b, and c, and the line AB.

Required to divide AB into parts proportional to a, b, and c. Construction. Draw AC, making A about 30°. From A on AC lay off AR, RL, and LH equal respectively to a, b, and c. Draw BH, and through R and L construct parallels to BH, cutting AB at K and G respectively. Then AK, KG, and GB are the required parts of the line AB.

The proof is left to the student.

Construction 4

307. Upon a given side homologous to a side of a given polygon construct a polygon similar to the given one.

E

E

B

Given the polygon ABCDE and the line A'B'.

B'

Required to construct a polygon similar to ABCDE with AB and A'B' corresponding sides.

Construction (outline of method only). Draw all the diagonals from A. Then construct 2 = Z1, Z B' = ZB, and produce the sides of 2 and ▲ B' until they meet in C'.

Then construct 43, 6=25, etc.

The remainder of the construction and the proof are left to the student.

CONSTRUCTIONS INVOLVING PROPORTION

211. Given the perimeter, construct a triangle similar to a given triangle.

212. Given the perimeter, construct a rectangle similar to a given rectangle.

213. Divide one side of a triangle into two parts proportional to the other two sides.

214. Construct a triangle, given two sides, a and b, and the ratio of a to the third side c, expressed by two other given lines, h and k.

HINT. Find a fourth proportional to h, k, and a.

215. Construct a triangle, given one side, a, the ratio of a to b, and the ratio of a to č.

216. Draw a line parallel to one side of a rectangle, cutting off a rectangle similar to the given one.

217. Inscribe in a given circle a triangle similar to a given one. HINT. Circumscribe a circle about the given triangle. Join its center to the vertices and study the central angles so formed.

218. Circumscribe about a given circle a triangle similar to a given one.

219. Given one altitude, construct a triangle similar to a given

one.

220. Construct a trapezoid similar to but not equal to a given

one.

BOOK IV

SURFACE MEASUREMENT. AREAS

308. Congruence and equality. It follows from the definition of congruence (§ 24) that congruent figures are equal.

Suppose two plane figures are composed of, or may be divided into, the same number of parts. Then if for each part of the first there is one part of the second congruent to it, and only one, the two figures are equal. Hence it follows that two equal figures are not always congruent.

Thus the rectangle A and the right triangle B may be put together to form the trapezoid or the pentagon on the right, and these two polygons are equal but not congruent.

B

A

A

B

309. Unit of surface. Comparison of the size of plane figures requires measurement of length, since the unit of surface is a square whose side is the unit of length. This square is called the unit square. The unit of surface in practical use is the square inch or the square centimeter, or some multiple of them, as the square yard or the square meter respectively.

310. Area. The area of a plane figure is the number which expresses the ratio between its surface and the surface of the unit square.

It is evident that any closed plane surface, whether bounded by straight lines or by curves or in part by both, has an area. To find that area may be easy or it may be difficult. An important

214

part of the work of plane geometry is finding the area of the simple plane figures.

311. Area of a rectangle. The plane figure the measurement of whose surface most clearly illustrates the notion of area is a rectangle whose sides are exact multiples of the unit of length. Thus, rectangle ABCD five units long and three units wide. It can, by parallels to the sides, be divided into fifteen squares each equal to the unit square K. We say the area of the rectangle is fifteen square units.

D

A

C

B

K

If the length of the base, AB, or that of the altitude, AD, of the rectangle is a mixed number like 51, the area can still be stated in terms of K and is as before the product of the base by the altitude. This is still true even if the base and the altitude are both mixed numbers.

When the base or the altitude of the rectangle, or both, and the side of the square have no common unit of measure, the area of the rectangle can still be expressed in terms of K and is equal to the product of the base and altitude.

The proof of this last is difficult. It will be omitted, and Theorem 1, of § 312, will be assumed as the basis of all work on area.

Theorem .1

312. The area of a rectangle is the product of its base and altitude.

It should be noted that the product of two lines always means the product of their numerical measures.

313. Corollary 1. The area of a square is the square of one of its sides.

314. Corollary 2. The area of a right triangle is one half the product of the two sides about the right angle.