THEOREMS FOR DEMONSTRATION. 1. THE diagonals of a rectangle are equal to each other. 2. Two squares are to each other as the squares of their diagonals. 3. Two similar parallelograms are to each other as the squares of their corresponding diagonals. 4. A line drawn parallel to the parallel sides of a trapezoid divides the other sides proportionally. 5. Any two altitudes of a triangle are inversely proportional to the bases on which they fall. 6. The straight line joining the middle points of the sides of any triangle is parallel to the base of the triangle and equal to one-half of the base. 7. The triangle formed by joining the middle points of the sides of any triangle is similar to the given triangle and equivalent to one fourth of the triangle. 8. Any straight line drawn through the point of intersection of the diagonals of a parallelogram bisects the parallelogram. 9. If one of the parallel sides of a trapezoid is double the other, the diagonals will trisect each other. 10. The area of an equilateral triangle whose side is 1 is equal to one-fourth of the square root of 3. 11. In a right triangle the square of either of the two sides containing the right angle is equal to the rectangle contained by the sum and difference of the other sides. 12. In the right triangle ABC, right angled at B, if a line CD is drawn cutting AB in D, then AC2 +DB2 = AB2 + DC2. 13. If two right triangles have one leg equal, the sum of the other sides in one triangle is to the sum of the corresponding sides in the other as their difference is to the difference of the former. 14. In any right triangle the product of the legs is equal to the product of the hypotenuse into the perpendicular to the hypotenuse from the vertex of the right angle. 15. In a right triangle whose legs are equal, the square of the hypotenuse equals four times the area of the triangle. 16. If upon the sides of a right triangle as homologous sides, sim ilar polygons are constructed, the polygon on the hypotenuse is equivalent to the sum of the polygons on the other two sides. 17. The triangle formed by joining the middle points of two adjacent sides of a parallelogram is equivalent to one-eighth of the parallelogram. 18. In any triangle, if a perpendicular is drawn from the vertex to the base, the difference of the squares of the sides is equal to the difference of the squares of the segments of the base. 19. If one side of a right triangle is double the other, the perpendicular from the vertex upon the hypotenuse divides the hypotenuse into parts which are to each other as 1 to 4. 20. If two isosceles triangles have their legs equal each to each, and the altitude of one equal to half the base of the other, the trian gles are equivalent. 21. The area of a trapezoid is equal to the product of one of its legs into the distance from this leg to the middle point of the other leg of the trapezoid. 22. If from any point within a parallelogram lines are drawn to the four vertices, the sum of either pair of triangles having parallel bases is equivalent to one-half of the parallelogram. 23. The difference between the hypotenuse of a right triangle and the sum of the other two sides is equal to the diameter of the inscribed circle. PRACTICAL EXAMPLES. AREAS AND RIGHT TRIANGLES. 1. Required the perimeter and area of a rectangle whose sides are 16 inches and 20 inches respectively. 2. Required the area of a parallelogram whose base is 24 inches and altitude 18 inches. 3. Required the area of a triangle whose base is 25 inches and altitude 20 inches. 4. Required the area of a trapezoid whose altitude is 6 inches and the parallel sides 12 inches and 8 inches respectively. *5. Required the hypotenuse of a right triangle, the two legs be ing 6 and 8 inches respectively. Ans. 10 in. 6. Required the diagonal of a square whose sides are 6 inches. The diagonal of a rectangle whose sides are 5 and 12 inches respectively. Ans. 6 1/2 in.; 13 in. 7. Required the area of a lot in the form of hypotenuse of which is 91 ft., and one leg 35 ft. a right triangle, the Ans. 1631 sq. yd. 8. A ladder 78 feet long leans against a house, so that its foot is 30 feet from the house; how high does it reach? Ans. 72 ft. 9. A pole broke 75 ft. from the top, and fell so that the end struck 60 ft. from its foot; required the length of the pole. Ans. 120 ft. 10. There is a rectangular field whose sides are 64 rods and 25 rods respectively; required the side of a square field of the same Ans. 40 rods. area. 11. The perimeter of a rectangle is 216 feet, and the length is twice the breadth; required the area. Ans. 288 sq. yd. 12. A square and a rectangle have the same perimeter, 144 feet, and the length of the rectangle is 3 times its breadth; compare their areas. Ans. 4: 3. 13. A ladder whose length is 50 feet stands close up against a building; how far must its foot be drawn away from the building that the top may be lowered 10 feet? Ans. 30 ft. 14. Required the area of a lot in the form of an isosceles triangle whose base is 16 rods and each leg 10 rods. Ans. 48 sq. rd. 15. A lady owns a lot in the form of an equilateral triangle, each side being 12 rods; required its area. Ans. 62.35 sq. rd. 16. The bases of a trapezoid are 20 feet and 14 feet, and each leg is 5 feet; what is the area of the trapezoid? Ans. 68 sq. ft. 17. Find the side of a square whose area is equal to the area of two squares whose sides are 6 rods and 8 rods respectively. Ans. 10 rods. 18. Find the side of a square whose area equals the difference between the areas of two squares whose sides are 13 feet and 5 feet respectively. Ans. 12 ft. *The numbers 3, 4, and 5 are the smallest integers that can express the relation of the three sides of a right triangle. Another integral relation of the sides is 5, 12, and 13. 19. The area of a rhombus is 48 sq. ft., and the length of one diagonal is 8 feet; what is the other diagonal? Ans. 12 feet. 20. The altitude of a trapezoid is 8 feet and the bases are 10 feet and 12 feet respectively; required the base of an equivalent rectangle with an equal altitude. Ans. 11 ft. 21. If the linear scale of a map is 1 inch to 10 miles, how many square miles will a square inch on the map represent? 22. A man has a field in the form of a rectangle which contains 90 acres; what are its dimensions if its length is equal to twice its breadth ? Ans. 84.85 rd.; 169.70 rd. 23. A cemetery containing 135 acres is laid out in the form of a rectangle, its length being three times its breadth; required its length and breadth. Ans. 84.85 rd. ; 254.55 rd. 24. Given the two sides of a triangle 13 and 15 inches respectively, and the altitude on the third side 12 inches; required the third side. Ans. 14 in. 25. Given the three sides of a triangle 13 in., 14 in., and 15 in. respectively; required the area of the triangle. Ans. 84 sq. in. 26. A general wishing to draw up his troops in the form of a square, found by the first trial he had 100 men over; he then increased the side of the square by 2 men and found he lacked 136 men to complete the square; how many men had he in the corps? Ans. 3464. PRACTICAL EXAMPLES. RELATION OF LINES AND SURFACES. 1. If the sides of a triangle are respectively 3, 4, and 6, is the angle opposite 6 acute, right, or obtuse? 2. If the sides of a triangle are respectively 10, 24, and 26, what kind of angle is opposite 26? 3. If the sides of a triangle are respectively 5, 10, and 12, what is the kind of triangle? 4. The sides of a triangle are 8, 12, and 16, and in a similar triangle the side homologous to 8 is 14; required the other two sides. Ans. 21 and 28. 5. The sides of a triangle are 4, 5, and 6; what are the sides of a similar triangle 9 times as large? Ans. 12, 15, 18. 6. The sides of two similar triangles are 3, 4, 5, and 12, 16, 20; how many times the first triangle equals the second? 16 times. 7. The corresponding sides of two similar polygons are 6 and 30; the second polygon is how many times the first? 25 times. 8. In the triangle ABC, AB= 21, AC = 12, BC= 16; find the segments of AB formed by the bisector of C. Ans. 9; 12. 9. In an obtuse triangle ABC, having given AC = 8, AB = 12, and the projection of AC on AB produced equal to 2; find BC opposite the obtuse angle A. Ans. BC= 16. 10. In an acute triangle ABC, having given AB = 12, BC = 10, and the projection of BC on AB equal to 2, find AC opposite the acute angle B. Ans. AC 14. 11. Required the altitude of an obtuse triangle whose sides are respectively 14 and 10 inches, and base 6 inches. Ans. 5V/3. 12. Required the altitude and the area of an acute triangle whose base AB = 12 inches, and the other two sides 8 inches and 10 inches respectively. Ans. Alt. V7; area 15 1/7. 13. Required the length of the median CD of the triangle ABC, if the side AB = 10, AC = 12, and BC= 8. Ans. 179. 14. Given the two adjacent sides of a parallelogram 6 inches and 8 inches, and one diagonal 12 inches; to find the other diagonal (Th. XIV.). Ans. 2 1/14, or 7.48. 15. In a triangle, the side AC 18 in. and the side BC= 24 in., and a line parallel to the base AB cuts the side AC 6 in. from the vertex; how far from C does it cut the side BC? Ans. 8 in. 16. In a triangle ABC, BC= 24 in., AB= 21 in., and AC = 15 in.; find the parts into which BC is divided by the bisector of the angle A. Ans. 10; 14. 17. In a triangle ABC, AB = 24 in., BC = 12 in., and AC=15 in.; find the distance from the vertex of the angle B at which the bisector of the exterior angle cuts BC produced. Ans. 32 in. 18. In a circle two chords AB and CD intersect; the two seg ments of AB are 6 and 8, and one segment of CD is 9; what is the other segment of CD? Ans. 5. |