393. A plane figure bounded by three straight lines is a triangle. 394. The base of a triangle is the side on which it is sup(Any side of a triangle may be taken for its base.) posed to rest. 395. The altitude of a triangle is the perpendicular distance from the angle opposite the base, to the base, or to the base produced. (Produced- -con- 396. Triangles take different names, according to the relations of their sides. If the sides of a triangle are equal, it is an eqilateral triangle. If only two sides of a triangle are equal, it is an isosceles triangle. If no two sides are equal, it is a scalene triangle. If one of the angles is a right angle, it is a right-angled triangle, or a right triangle. ̧ 44440 Equilateral Isosceles Triangle. Scalene Triangle. Right-angled 397. Parallel lines are straight lines that have the same direction but do not coincide, and can never meet, however far they may be produced. 398. A plane figure bounded by four straight lines is a quadrilateral. (Quadrilateral means four-sided.) 399. Quadrilaterals take different names from their angles and from the relation of the sides to each other. 400. A quadrilateral which has no two sides parallel is a trapezium. 401. A quadrilateral which has only two sides parallel is a trapezoid. 402. A quadrilateral that has its opposite sides parallel is a parallelogram. 403. Parallelograms take different names from their angles and the relation of the sides to each other. 404. A parallelogram that has all its angles right angles, and all its sides equal, is a square. 405. A parallelogram that has all its angles right angles, and only its opposite sides equal, is called a rectangle. 406. A parallelogram that has its sides all equal, but whose angles are not right angles, is called a rhombus. 407. A parallelogram that has only its opposite sides equal, and whose angles are not right angles, is called a rhomboid. 408. A straight line that joins the vertices of two angles, not adjacent, is a diagonal. - To find the Areas of Quadrilaterals and Triangles. The Rectangle, including the Square. We have already found that to compute the area of a rectangle, we must multiply the number of the proposed square units of measure which can be placed on one side of the rectangle by the number of corresponding linear units in the adjacent side. Example.-Let the figure represent a rectangle 14 inches wide and 19 inches long. What is the area? Solution.-19 square inches can be placed on the side c d, and, since there are 14 such rows, there will be 14 times 19 sq. in. in the whole rectangle 266 sq. in. a d C The Rhomboid and Rhombus.-Example.-Let it be required to compute the area of a rhomboid, 10 in. long and 6 in. wide. Solution.-6 x 10 □ in. = 60 □ in. Ans. If from either end of a rhomboid we cut a right-angled triangle, and add it to the other end, as indicated by dotted lines in the figure, we should form a rectangle equivalent to the rhomboid; hence, 409. To find the area of a rhomboid: Multiply the length of one of two parallel sides by the distance between them. The rule for the rhombus is the same. It is to be observed that, to obtain the width of a rhombus or rhomboid, we do not measure a side, but the perpendicular distance between parallel sides. The Triangle.—Example.-Given the base of a triangle 14 yd. and the altitude 9 yd., to find the area. The triangle is one half of a parallelogram having the same base and altitude, as may be scen by the above diagram. Hence, 410. To find the area of a triangle: Find the area of a rectangle of the same base and altitude, and take one half of it. 411. The following rule is sometimes necessary : When the three sides of a triangle are given, to find the area: From half the sum of the three sides subtract each side separately. Multiply the half sum and the three remainders together, and extract the square root of the product. The Trapezoid. - Example. Given the length of each of the two parallel sides of a trapezoid, 6 and 10 feet, and the distance between them, 5 feet, to find the area. Solution.-6 ft.+ 10 ft. 16 ft. 1/2 of 16 ft. 8 ft. 5 x 8 sq. ft. = 40 sq. ft. Ans. By inspection of the figure it will be seen that, by the aid of dotted lines, we have constructed a rhomboid equal in area to the trapezoid whose area is required; and, further, it is plain that the side of this rhomboid is equal to half the sum of the two parallel sides of the trapezoid. Hence, 412. To find the area of a trapezoid: Multiply one half the sum of the parallel sides by the distance between them. 413. The Trapezium.-The surface of a trapezium may be found by dividing it into two triangles; then having measured the length of the diagonal and the two perpendiculars, we calculate the area of each triangle separately. The sum of the areas of the two triangles is the area of the trapezium. Example. In a trapezium A B C D, we measure the diagonal A C, and find it to be 24 feet; also the perpendiculars, and find one to be 18, the other 9 feet. What is the area of the trapezium? Solution. A B Area of the triangle A B C = 24 x 9216 sq. ft., 1/1⁄2 of 216 = 108 sq. ft. Area of the triangle A D C = 24 × 18 = 432 sq ft., 1/2 of 432 = 216 sq. ft. Area of ABC + ADC, or the whole tm.= 648 sq. ft., 1/2 =324 sq. ft. Ans. 1. How many acres in a piece of woodland 220 yd. in length and 1 furlong in width? 2. How many square miles in a township 5 miles and 40 chains square? 3. How many square feet in a floor 20 ft. long and 5 yd. wide ? 4. Find the surface of a pane of glass measuring 371⁄2 in. long and 23 in. wide. 5. How many square yards in the four walls of a room 15 ft. 6 in. high and 80 ft. in compass? 6. A rectangular pavement, 50 ft. 9 in. long and 12 ft. 6 in. wide, was laid with a central line of stone 5 ft. wide at $1.75 a running foot; the sides were flanked with brick at 80¢ per square yard. What did the paving cost? 7. How many square feet in a surface 24 ft. long 20 ft. wide? How many in another surface of half these dimensions? 8. I have a box without a lid; it is 5 ft. long, 4 ft. wide, and 3 ft. deep, interior dimensions. How many square feet of zinc will it take to line the bottom and sides of this box? 9. Find the area of a rhomboid whose length is 1 yd. 2 ft. 6 in., and whose width is 2 ft. 3 in. Draw this figure on your slate, with the scale reduced by 12. 10. What is the height of a rhomboid whose area is 12 A. and its length 13 chains? 11. The four eaves of a pyramidal roof measure each 44 ft. 3 in., and the common peak of the four triangles has a perpendicular distance of 24 ft. from the eaves. What is the area in slaters' squares (1) of one triangle? (2) of the roof? 12. I have a triangular garden containing 2333 square yards. The perpendicular distance from the apex to the base is 20 ft. What is the length of the base? 13. A triangular field, whose sides are unequal, contains 5 The base-line measures 1/4 mile. What is the altitude in acres. chains? 14. What is the area of a triangle whose three sides are 13, 14, and 15 ft.? 15. What is the area in acres of a triangular field whose three sides measure respectively 47, 58, and 69 rods? 16. The parallel sides of a trapezoid measure respectively 3 ft. and 6 ft. 8 in.; the perpendicular distance between them is 2 ft. What is the area? 17. Find the area of a trapezium whose diagonal is 168, and one perpendicular 42, the other 56. 18. How many centares in a rhomboid one side of which measures 50 meters, the perpendicular distance to the opposite side being 35 meters? 19. What is the area of a square field, the diagonal of which measures 174 meters ? |