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When one straight line meets another, making the adjacent angles equal, the lines are perpendicular to each other, and the angles are Right Angles.
An Acute Angle is an angle less than a right angle. An Obtuse Angle is an angle greater than a right angle.
238. A Surface is that which has length and breadth only.
A Plane Surface, or a Plane, is a surface in which, if any two points are taken, the straight line which joins these points will lie wholly in the surface.
239. A Plane Figure is a plane surface bounded by lines either straight or curved.
240. A Polygon is a plane figure bounded by straight lines.
241. A Polygon of three sides is called a Triangle; one of four sides, a Quadrilateral; one of five sides, a Pentagon; one of six sides, a Hexagon, etc.
The lines A B, BC, CA; DE, EF, FG, GD; and HI, IJ, JK, KL, LH, of the above polygons are called the sides, and in any polygon the sum of the sides is called the perimeter.
The diagonal of a polygon is a line joining any two vertices not consecutive.
The area of a plane figure is the number of square units it contains.
242. A Triangle is a polygon of three sides ; as, A BC.
The base of a triangle is the side
which it seems to stand ; as, A B. The point opposite the base is called the vertex; as, C.
The altitude of a triangle is the perpendicular drawn from
A the vertex to the base; as, C D.
Triangle. 243. A Parallelogram is a quadrilateral whose opposite sides are parallel; as, ABCD.
The base of a parallelogram is the side upon which it seems to stand; as, A B. The altitude is the perpendicular dis- A tance between the base and
Parallelogram. the side opposite; as, D E.
A Rectangle is a parallelogram whose angles are all right angles; as, ABCD.
A Trapezoid is a quadrilateral two of whose sides are parallel ; as, MNOP. The two parallel sides are called the bases of the trapezoid ; as, M N and PO. The altitude is the perpendicular dis
Trapezoid. tance between the bases; as, P Q.
AREA OF POLYGONS.
244. The Rectangle.
Let the figure A B C D represent a rectangle 8 inches long and 4 inches wide. What is its area ?
If we divide the base and altitude into inches, and draw lines through these points of division parallel to the sides, the rectangle will be divided into equal squares.
There will be 8 square inches in one row; and since there are 4 such rows, there will be 4 times 8 square inches, or 32 square inches.
From this we derive the
To find the area of a rectangle, multiply the base by the altitude.
1. What is the area of a rectangle 20 ft. long and 16 ft. wide ?
2. What is the area of a rectangle 12 ch. long and 5 ch. wide ?
3. How many square yards in a rectangle 40 ft. long and 18 ft. wide ?
4. How many acres in a rectangle 40 rd. long and 12 rd. wide ?
5. How many acres in a rectangle 36 chains long and 10 chains wide ?
6. Find the area of a square whose sides are each 8 ft. 6 in.
7. Find the length in rods of a rectangular field whose area is 16 acres and width 40 rd.
8. A rectangular yard 40 ft. long and 30 ft. wide is surrounded by a walk 4 ft. wide; how many square feet in the walk?
9. If a rectangular field is 80 rd. long and 60 rd. wide, how much is the field worth, at $140 an acre ?
10. A city lot is 5 rd. wide and 18 rd. long; what is it worth, at $1600 an acre ?
11. There is a hollow square whose outer side is 120 rd. and the inner side 40 rd.; how many acres does it contain ?
12. A rectangular field is 120 rd. long and 40 rd. wide; how many more acres will a square field of equal perimeter contain ?
13. How many more feet in two sides of a square acre than in its diagonal ?
14. There is a rectangular field whose sides are 125 rd. and 20 rd., respectively; what is the side of a square field of equal area?
245. The Parallelogram.
If the rectangle (Fig. 1) be divided by the line A E, and the triangle A D E be placed at the right of the figure, we shall have a parallelogram of the form ABDE (Fig. 2).
It is evident that the base, the altitude, and the area remain
To find the area of a parallelogram, multiply the base by the altitude.
WRITTEN EXERCISES. 1. What is the area of a parallelogram whose base is 32 in. and altitude 25 in. ?
2. How many acres in a parallelogram whose base is 120 rd. and altitude 15 rd.?
3. How many acres in a farm 320 rd. long and 45 rd. wide ?
4. How many acres in a parallelogram 60 ch. long and 45 ch, wide ?
246. The Triangle.
If the parallelogram (Fig. 2, ART. 245) be divided by a line drawn from B to E, it will form two equal triangles. It is evident that the width and altitude of each triangle are the same as the width and altitude of the parallelogram, but the area of each triangle is only one-half of the area of the parallelogram. Hence the
RULE. To find the area of a triangle, multiply the base by half the altitude.
WRITTEN EXERCISES. 1. Find the area of a triangle whose base is 120 ft. and altitude 60 ft.
2. How many acres in a triangular field whose base is 80 rd. and altitude 60 rd.?
3. How many acres in a triangular field whose base is 40 ch. and altitude 30 ch.?
4. What is the altitude of a triangle whose base is 360 ft. and area 1 acre?
NOTE.-In Geometry it is proved that if the three sides of a triangle are given and not the altitude, the area can be found as follows: