611. EXERCISE. lower bases are 7 ft. The area of a trapezoid is 12 sq. ft. The upper and and 5 ft. respectively. Find its altitude. 612. EXERCISE. The area of a trapezoid is 24 sq. in. The altitude is 4 in., and one of its parallel sides is 7 in. What is the other parallel side ? PROPOSITION VIII. THEOREM 613. Triangles that have an angle in one equal to an angle in the other, are to each other as the products of the including sides. Prove Proof. Lay off BG = ED and BH = EF. Draw GH and AH. ABHG ᏴᏀ BH 614. EXERCISE. Prove § 613, using this pair of triangles. 615. EXERCISE. The triangle ABC has B equal to E of triangle DEF. The area of ABC is double that of DEF. AB is 8 ft., BC is 6 ft., and DE is 12 ft. How long is EF? 616. Similar triangles are to each other as the squares of their homologous sides. 617. EXERCISE. Similar triangles are to each other as the squares of their homologous altitudes. 618. EXERCISE. In the triangle ABC, ED is parallel to AC, and CD = 1 DB. How do the areas of triangles ABC and BDE compare? B E 619. EXERCISE. The side of an equilateral triangle is the radius of a circle. A The side of another equilateral triangle is the diameter of the same circle. How do the areas of these triangles compare? 620. EXERCISE. Two similar triangles have homologous sides 12 ft. and 13 ft. respectively. Find the homologous side of a similar triangle equivalent to their difference. 621. EXERCISE. The homologous sides of two similar triangles are 3 ft. and 1 ft. respectively. How do their areas compare? 622. EXERCISE. Similar triangles are to each other as the squares of any two homologous medians. 623. EXERCISE. The base of a triangle is 32 ft., and its altitude is 20 ft. What is the area of a triangle cut off by drawing a line parallel to the base at a distance of 15 ft. from the base? 624. EXERCISE. A line is drawn parallel to the base of a triangle dividing the triangle into two equivalent portions. In what ratio does the line divide the other sides of the triangle ? 625. EXERCISE. Draw a line parallel to the base of a triangle, and cutting off a triangle that shall be equivalent to one third of the remaining portion. 626. EXERCISE. Equilateral triangles are constructed on the sides of a right-angled triangle as bases. If one of the acute angles of the rightangled triangle is 30°, how do the largest and smallest equilateral triangles compare in area? 627. EXERCISE. In the triangle ABC, the altitudes to the sides AB and AC are 3 in. and 4 in. respectively. Equilateral triangles are con structed on the sides AB and AC as bases. Compare their areas. 628. EXERCISE. The homologous altitudes of two similar triangles are 5 ft. and 12 ft. respectively. Find the homologous altitude of a triangle similar to each of them and equivalent to their sum. 629. EXERCISE. Draw a line parallel to the base of a triangle, and cutting off a triangle that is equivalent to of the remaining trapezoid. 630. EXERCISE. Through O, the point of intersection of the altitudes of the equilateral triangle ABC, lines are drawn parallel to the sides AB and BC respectively and meeting AC at x and y. Compare the areas of triangles ABC and Oxy. 631. Similar polygons are to each other as the squares of their homologous sides. Proof. From the vertex A draw all the possible diagonals. From F, homologous with 4, draw the diagonals in FGHIJ. 632. COROLLARY I. Similar polygons are to each other as the squares of their homologous diagonals. 633. COROLLARY II. In similar polygons homologous triangles are like parts of the polygons. [This was shown in the proof of the proposition.] 634. EXERCISE. The area of a certain polygon is 24 times the area of a similar polygon. A side of the first is 3 ft. 635. EXERCISE. The homologous sides of two similar polygons are 8 in. and 15 in. respectively. Find the homologous side of a similar polygon equivalent to their sum. 636. EXERCISE. The areas of two similar pentagons are 18 sq. yds. and 25 sq. yds. respectively. A triangle of the former pentagon contains 4 sq. yds. What is the area of the homologous triangle in the second pentagon ? 637. EXERCISE. If the triangle ADE [see figure of § 631] contains 12 sq. in., and triangle FIJ contains 9 sq. in., how do the areas of ABCDE and FGHIJ compare? 638. EXERCISE. The homologous diagonals of two similar polygons are 8 in. and 10 in. respectively. Find the homologous diagonal of a similar polygon equivalent to their difference. 639. EXERCISE. Connect C with m, the middie point of AD, and H with n, the middle point of FI [see figure of § 631], and prove 640. EXERCISE. ratio of their sides? If one square is double another square, what is the 641. EXERCISE. Construct a hexagon similar to a given hexagon and equivalent to one quarter of the given hexagon. 642. EXERCISE. Construct a square equivalent to of a given square. |