280. Since a parallelogram is equivalent to a rectangle of the same base and height, it follows that the ratio of the surfaces of two parallelograms is that of the

products obtained by multiplying the number of units of length in the base by the number in height. Thus the ratio of the parallelograms represented in figure 279 24 X IO is equal to

18 X 15

The ratio of the surfaces of two triangles is also equal to that of the products of the base and height.

281. When the triangles are similar the ratio of their altitudes is equal to that of their bases; hence the ratio of the surfaces of similar triangles is equal to the ratio of the squares of their homologous sides.

Rectangles, parallelograms, or triangles, are equivalent when in multiplying the base of each by its altitude we obtain the same result.

On a given base to place a rectangle equivalent to a given rectangle.

282. Let A B be the given base and A D, AF the sides of the given rectangle (fig. 280). If x the height of the " required rectangle, AB X x= ADXAF; hence x is a fourth F proportional to A B, AD, and AF, so that the construction is that indicated in the figure. A It will be useful to compare

B

D

Fig. 280.

the theorems of equivalence of triangles with those of

similarity.

Equivalence.

Two triangles are equivalent

Ist. When three sides of the one are equal to three sides of the other, each to each.

2nd. When two sides and the included angle of one are respectively equal to two sides and included angle of the other.

3rd. When two angles and the included side of one are respectively equal to two angles and included side of the other.

4th. When the triangles are right-angled, and the hypothenuse and a side. of one are respectively equal to the hypothenuse and side of the other.

Similarity.

Two triangles are simi

lar

Ist. When the sides of one are proportional to those of the other.

2nd. When two sides of the one are proportional to two sides of the other, and the included angles are equal.

3rd. When two angles of the one are equal each to each to two angles of the other.

4th. When the triangles are right-angled, and the ratio of the hypothenuse to a side of the one is equal to the corresponding ratio of the other.

Questions for Examination.

1. Define similar figures, ratio of similitude, and homologous sides.

2. Prove that two triangles are similar when two angles of one are equal to two angles of the other, each to each.

3. Two triangles which have their sides perpendicular or parallel, each to each, are similar.

4. Two triangles which have their sides proportional are similar.

5. Two triangles which have two sides of the one proportional to two sides of the other, and the included angles equal, are similar.

6. On a given straight line to construct a triangle similar to a given triangle.

7. If in a right-angled triangle a perpendicular be drawn from the right-angle to the hypothenuse, it will divide the triangle into two others similar to the whole and to one another.

8. If from a point in the diagonal of a parallelogram straight lines be drawn parallel with the sides, two parallelograms will be formed similar to the whole and to one another.

Theorems and Problems.

1. Construct a parallelogram on a given base similar to a given parallelogram.

2. Construct a triangle having given the three altitudes.

3. In a given triangle, place a parallelogram similar to a given parallelogram.

4. In a given circle inscribe a triangle similar to a given triangle.

5. On a given base place a triangle equivalent to a given triangle.

6. Construct an isosceles triangle equivalent to a given triangle, and having a given vertical angle.

7. If two triangles, A EF, AB C, have a common angle A, then ABC AEF:: ABX AC: AEX AF.

8. Determine two straight lines such that the sum of their squares may equal a given square, and their rectangle equal a given rectangle.

9. Divide a given arc of a circle into two arcs which shall have their chord in a given ratio.

10. On a given straight line describe a rectilineal figure similar to a given rectilineal figure.

II. If two systems of parallel straight lines at equal distances cut one another, the figures between them will be all similar parallelograms.

12. ABC is a triangle. Draw a straight line AD meeting the base, or the base produced at D, so that ABD and ABČ may be a similar triangle.

13. Draw a straight line such that the perpendiculars let fall from any point in it on two given lines may be a given ratio.

14. It is required to divide a given finite straight line into two parts, the squares of which shall have a given ratio to each other.

15. From the obtuse angle of a triangle, it is required to draw a line to the base, which shall be a mean proportional between the segments of the base. How many answers does this question admit of?

16. To draw a line from the vertex of a triangle to the base,

which shall be a mean proportional between the whole base and one segment.

17. Any two triangles being given, to draw a straight line parallel to the side of the greater, which shall cut off a triangle equal to the less.

18. In any right-angled triangle ABC (whose hypothenuse is A B) bisect the angle A by AD meeting C B in D, and prove that 2 A C2: A C2-C D2 :: BC: CD.

19. On two given straight lines similar triangles are described. Required to find a third, on which, if a triangle similar to them be described, its area shall equal the difference of their areas.

20. If a straight line be divided into any two parts, find the locus of the point at which these parts subtend equal angles. 21. It is required to bisect a triangle by a line drawn parallel to the base.

22. Find three points in the sides of a triangle, such that when they are joined, the triangle shall be divided into four equal triangles.

23. From a given point in the side of a triangle, to draw lines to the sides which shall divide the triangle into any number of equal parts.

24. Describe a rectangular parallelogram which shall be equal to a given square, and have its sides in a given ratio.

25. It is required to cut off from a rectangle a similar rectangle which shall be any required part of it.

26. The diagonals of a trapezoid, cut one another in the same ratio.

Arithmetical Exercises.

1. The area of a triangle is 135 sq. yds., and its base is 13 yds. long: find the area of a similar triangle on a base 8 yds. long. Ans. 51 124 yds.

2. A parallelogram contains 36 sq. ft.: find the areas of the parallelograms formed by drawing lines through a point in the diagonal, dividing it into parts in the ratio 4: I.

Ans. 34 56 and 1.44 sq. ft.

3. A rectangular field, 470 links long and 260 broad, is to be laid out in two square grass plots surrounded by a walk of the same width throughout. Give a construction to find the breadth of the walk, and find it. Ans. 50 links.

4. What is the ratio of similar triangles whose altitudes are respectively 14 ft. and 5 ft. 6 in. ? Ans. 196 394. 5. What is the ratio of similar parallelograms whose diagonals are respectively 15 in. and II in.?

6. In two semicircles triangles are inscribed, one angle in each being double of another. The area of one triangle is 36 × √3 sq.

ft., and that of the other, 12×3 sq. ft.: find the radii of the semicircles. Ans. 3 ft. and 6 ft. 7. The areas of two equilateral triangles are respectively 61 sq. yds. and 7 sq. yds. 1 sq. ft. : find the ratio of their sides.

Ans. 1.

CHAPTER XII.

POLYGONS.

A plane figure bounded by straight lines is called a polygon. Some polygons have special names.

A polygon of three sides is termed a triangle.

quadrilateral.
pentagon.
hexagon.

heptagon.
octagon.
decagon.
dodecagon.

It is immaterial whether we name the polygon from the number of its sides or the number of its angles, for it is evident that every polygon has as many sides as angles.

If lines be drawn from any one of the angular points of a polygon to all the others, the polygon is divided into as many triangles less two as it has sides. 283. Let there be for example, a polygon of seven sides (fig. 281). If one apex be joined to all the others, we have six lines, which form between them, at the apex, (7-2) angles, and divide the polygon into (7-2) corresponding triangles. From one apex (7-3) diagonals are drawn, and so from all the angular points seven times this number may be drawn.

Fig. 281.

It