Solution. Upon BC as a diameter describe a semicircumference. At one extremity of BC, as B, erect a perpendicular to BC, as DB, equal to a side of 4. Draw DE || BC meeting the semicirDraw EF || DB meeting BC in F. cumference in E. With base CF and altitude BF construct rectangle H. Then, H is the required rectangle. Q.E.F. 371. Problem. To construct a rectangle equivalent to a given square, and having the difference of its base and altitude equal to a given line. Data: Any square, as 4, and the line BC. Required to construct a rectangle equivalent to 4, and having the difference of its base and altitude equal to BC. Solution. On BC as a diameter describe a circumference. At one extremity of BC, as B, erect a perpendicular to BC, as BD, equal to a side of 4. Through 0, the center of the circle, draw DF intersecting the circumference in E and meeting it in F. Then, FD — ED = EF, or BC. With base FD and altitude ED construct rectangle H. Then, H is the required rectangle. Proof. By the student. SUGGESTION. Refer to §§ 315, 269. Q.E.F. Proposition XXIX 372. Problem. To construct a polygon similar to a given polygon and equivalent to any other given polygon. с A B H d Data: Any two polygons, as A and B. Required to construct a polygon similar to 4 and equivalent to B. Solution. Find c, the side of a square equivalent to 4, and d, the side of a square equivalent to B, and let e be a side of 4. Find a fourth proportional to c, d, and e, as f. Upon f homologous to e construct H similar to 4. Q.E.F. a. If it is the rectangle formed by the segments of one of two intersecting chords, to the rectangle formed by the segments of the other. § 357 b. If it is formed by a secant and its external segment, to a rectangle formed by another secant from the same point and its external segment. § 359 c. If it is formed by the two sides of a triangle, to the rectangle formed by the segments of the base, made by the bisector of the vertical angle, plus the square upon the bisector. § 360 d. If it is formed by two sides of a triangle, to the rectangle formed by the altitude upon the third side and the diameter of the circumscribing circle. 2. Rectangles are in proportion, a. If they have equal altitudes, to their bases. b. If they have equal bases, to their altitudes. c. To the products of their bases by their altitudes. $ 361 $ 327 § 328 $ 329 3. A parallelogram is equivalent, a. To the rectangle which has the same base and altitude. tude. b. To another parallelogram which has an equal base and an equal alti § 331 § 333 4. Parallelograms are in proportion, a. If they have equal altitudes, to their bases. b. If they have equal bases, to their altitudes. § 333 c. To the products of their bases by their altitudes. § 333 § 333 5. A triangle is equivalent, § 334 c. To the products of their bases by their altitudes. a. To one half the rectangle which has the same base and altitude. b. To another triangle which has an equal base and an equal altitude. § 336 c. To another triangle which has an angle equal to an angle of the first, and the products of the sides, including the equal angles, equal. 6. Triangles are in proportion, a. If they have equal altitudes, to their bases. b. If they have equal bases, to their altitudes. § 341 § 336 § 336 d. If they have an angle of one equal to an angle of the other, to the products of the sides including the equal angles. § 336 sides. e. If they are similar triangles, to the squares upon their homologous § 340 f. If they are similar triangles, to the squares upon any of their homologous lines. § 342 $ 343 7. A trapezoid is equivalent, a. To one half the rectangle which has the same altitude and a base equal to the sum of the parallel sides. § 337 8. The square upon a line is equivalent, a. If the line is the sum of two lines, to the sum of the squares upon the lines plus twice the rectangle formed by them. § 347 b. If the line is the difference of two lines, to the sum of the squares upon the lines minus twice the rectangle formed by them. $ 348 c. If the line is the hypotenuse of a right triangle, to the sum of the squares upon the other two sides. $ 349 ד. d. If the line is the side of an oblique triangle, opposite an acute angle, to the sum of the squares upon the other two sides minus twice the rectangle formed by one of those sides and the projection of the other upon that side. § 353 e. If the line is the side opposite an obtuse angle of a triangle, to the sum of the squares upon the other two sides plus twice the rectangle formed by one of those sides and the projection of the other upon that side. 9. The sum of two squares is equivalent, § 354 a. If they are the squares upon any two sides of an oblique triangle, to twice the square upon one half the third side plus twice the square upon the median to that side. 10. The difference of two squares is equivalent, § 355 a. If they are the squares upon any two sides of an oblique triangle, to twice the rectangle formed by the third side and the projection of the median upon that side. 11. Similar polygons are in proportion, a. To the squares upon their homologous sides. b. To the squares upon any of their homologous lines. 12. The area of a figure is equal, § 356 § 344 § 345 a. If it is a rectangle, to the product of its base by its altitude. b. If it is a parallelogram, to the product of its base by its altitude. c. If it is a triangle, to half the product of its base by its altitude. d. If it is a trapezoid, to half the product of its altitude by the sum of its parallel sides. § 330 § 332 § 335 § 338 SUPPLEMENTARY EXERCISES Ex. 538. The straight line joining the middle points of the parallel sides of a trapezoid bisects the trapezoid. Ex. 539. The lines joining the middle point of either diagonal of a quadrilateral to the opposite vertices divide the quadrilateral into two equivalent parts. Ex. 540. Two triangles are equivalent, if they have two sides of one respectively equal to two sides of the other, and if the included angles are supplementary. Ex. 541. O is any point on the diagonal AC of the parallelogram ABCD. If the lines DO and BO are drawn, prove that the triangles AOB and AOD are equivalent. Ex. 542. A rhombus and a rectangle have equal bases and equal areas. One side of the rhombus is 15m and the altitude of the rectangle is 12m. What are their perimeters? Ex. 543. The area of a rhombus is equal to one half the product of its diagonals. Ex. 544. The diagonals of a rhombus are 64 rd. and 37 rd. What is the area of the rhombus ? Ex. 545. The base of a triangle is 75m, and its altitude is 60m. Find the perimeter of an equivalent rhombus, if its altitude is 45m. Ex. 546. Find the area of a rhombus, if the sum of its diagonals is 12 in. and their ratio is 3: 5. Ex. 547. A man travels 25 miles east from a certain town, and another man travels 36 miles north from the same town. How far apart are the men? Ex. 548. The shortest side of a triangle acute-angled at the base is 45 ft. long, and the segments of the base made by a perpendicular from the vertex are 27 ft. and 77 ft. How long is the other side? Ex. 549. The sides of a triangle are 25m and 17m, and the lesser segment of the base made by a perpendicular from the vertex is 8m. What is the length of the base? Ex. 550. In a right triangle the base is 3dm, and the difference between the hypotenuse and perpendicular is 1dm. What are the hypotenuse and perpendicular? Ex. 551. In a right triangle the hypotenuse is 5dm, and the difference between the base and perpendicular is 1dm. Find the base and perpendicular. Ex. 552. The sides of a right triangle are in the ratio of 3, 4, and 5, and the perpendicular upon the hypotenuse from the vertex of the right angle is 20 yd. What is the area of the triangle? Ex. 553. If in any triangle a perpendicular is drawn from the vertex to the base, the difference of the squares upon the sides is equivalent to the difference of the squares upon the segments of the base. Ex. 554. In a right triangle the square on either side containing the right angle is equivalent to the rectangle contained by the sum and the difference of the other sides. Ex. 555. If the diagonals of a quadrilateral intersect at right angles, prove that the sum of the squares upon one pair of opposite sides is equivalent to the sum of the squares upon the other pair. Ex. 556. The altitude of an equilateral triangle is 60 in. How long are its sides? Ex. 557. Through D and E, the middle points of the sides AC and BC of the triangle ABC, any two parallel straight lines are drawn meeting AB or AB produced in the points F and G. Prove that the parallelogram DFGE is equivalent to half the triangle ABC. Ex. 558. The four triangles into which a parallelogram is divided by its diagonals are equivalent. |