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ART. 453.—Mensuration treats of the measurements of angles, lines, surfaces and volumes.
ART. 454.-A Line has length only.
ART. 455.—A Straight Line is one that does not change its direction : it is the shortest distance between two points.
ART. 456.—A Curved Line is one that changes its direction at every point.
ART. 457.–Parallel Lines are those that have the same direction.
ART. 458. An Angle is the opening between two straight lines that meet. Thus, the opening be
B tween the lines A B and C B is an angle.
ART. 459. — A Right Angle is an angle formed by one line perpendicular to another. Thus, the angles A B C and C B D are right angles.
ART. 460.–An Acute Angle is an angle less than a right angle; as D BC.
ART. 461.—An Obtuse Angle is greater than a right angle; as A BD.
ART. 462.-The Vertex of an angle is the point where the sides meet. ART. 463.-A Surface is that which has length and
breadth but no thickness : it may be either plane or curved.
ART. 464.-A Plane Surface is a surface such that if any two of its points be joined by a straight line, every part of the line will touch the surface.
Measurement of Lines and Surfaces.
ART. 465.—A Plane Figure is a figure all of whose parts are in the same plane.
ART. 466. — A Polygon is a portion of a plane bounded by straight lines. ART. 467.
1.—The Perimeter of a polygon is the sum of its sides, or the distance around it.
ART. 468.-The Diagonal of a polygon is a line connecting the vertices of two angles not adjoining each other.
ART. 469.-The Area of a plane figure is the number of square units in its surface.
ART. 470.-A Triangle is a polygon with three sides; a Quadrilateral is a polygon with four sides; a Pentagon is a polygon with five sides ; a Hexagon is a polygon with six sides ; etc.
THE TRIANGLE. A B C represents a triangle.
ART. 471.—The Base is the side on which it is supposed to stand.
ART. 472.—The Altitude is a perpendicular line drawn from the vertical
angle to the base : thus, C D is the altitude of the triangle A BC.
ART. 473.-An Equilateral Triangle is a triangle that has all its sides equal.
ART. 474.–An Isosceles Triangle is a triangle that has two of its sides equal.
ART. 475.—A Scalene Triangle is a triangle that has no two sides equal.
ART. 476.-A Right-Angled Trian
B gle is a triangle that has one right angle. Thus, A C B is a right-angled triangle, because the angle ACB is a right angle. The side A B, opposite the right angle is the hypotenuse, the side AC the base, and the side BC the altitude.
A right-angled triangle is equal to one half a rectangle having the same base and altitude.
ART. 477.-Rule for finding the area of any triangle. - Multiply the base by one half the altitude.
WRITTEN EXERCISES. 1. What is the area of a right-angled triangle whose base is 8 feet and altitude 6 feet ?
2. What is the area of a right-angled triangle whose base is 30 rods and altitude 20 rods ?
3. How many square feet in a triangle whose base is 40.5 feet and its altitude 30.25 feet ?
NOTE.—When the three sides are given, and not the altitude, from onehalf the sum of the sides subtract each side separately ; multiply half the sum and the remainders together, and extract the square root of the product.
4. What is the area of a triangle whose sides respectively are 30, 40 and 50 chains ?
5. The base of a right-angled triangle is 6 feet and the height 5 feet : what is the hypotenuse ?
NOTE.—It has been shown that the square of the hypotenuse equals the sum of the squares of the other two sides ; the difference between the square of the hypotenuse and the square of one of the sides is the square of the remaining side. Hence, when an two sides of a rightangled triangle are given, the third can be found.
6. If the hypotenuse is 10 feet and the altitude 8 feet, what is the base ?
7. If the hypotenuse is 9 feet and the base 8 feet, what is the altitude ?
8. A certain room is 60 feet long and 40 feet wide : what is the distance between the opposite corners ?
9. A ladder 32 feet long reached a window 24 feet above ground : how far was the base of the ladder from the side of the house ?
10. A flagstaff was broken 40 feet from the ground by the wind, and fell so that the top rested 30 feet from the foot of the staff : how long was the flagstaff before it was broken ?
11. A surveying expedition separated into two parties, one traveling north at the rate of 3 miles an hour and the other west at the rate of 31 miles an hour: how far were they from each other at the end of 12 hours ?
12. Illustrate by an original problem the measurement of triangles.
THE QUADRILATERAL. ART. 478.—A Quadrilateral is a polygon having four sides.
ART. 479.-A Parallelogram is a quadrilateral whose opposite sides are equal and parallel.
ART. 480.—When the parallelogram is right-angled, it is called a rectangle ; when the four sides are equal, it is a square; when not right-angled, it is a rhomboid; when the sides of the rhomboid are equal, it is a rhombus.
ART. 481.-A Trapezoid is a quadrilateral having two of its sides parallel.
ART. 482.-A Trapezium is a quadrilateral having no two of its sides parallel.
ART. 483.—The Altitude of a parallelogram or of a trapezoid is the perpendicular distance between its parallel sides.
ART. 484.—To find the area of a parallelogram.-Multiply the base by the altitude.
WRITTEN EXERCISES. 1. What is the area of a parallelogram 30 feet long and 16 feet wide ?
2. What is the difference between the areas of two lots, one of which is 250 rods long, 60 rods wide, and the other 140 rods long and 80 rods wide ?
3. Illustrate by an original problem the method of measuring parallelograms.
ART. 485.—To find the area of a trapezoid.—Multiply one half the sum of the parallel sides by the altitude.