Ex. 1042. In a given line AB, to find a point C such that AC: BC = 1: √2. Ex. 1043. To construct a parallelogram similar to a given parallelogram and having a given diagonal. Ex. 1044. To construct a triangle similar to a given triangle and having a given perimeter. Ex. 1045. If a and b are two given lines, construct a line equal to a2 b Ex. 1046. From a point without a circle, to draw a secant whose external segment is equal to one half the secant. Ex. 1047. To construct a circle, touching a given circle in a given point, and touching a given line. HINT. Draw a tangent to the circle. Ex. 1048. To construct a circle, touching two parallel lines and passing through a given point. Ex. 1049. In a given circle, to inscribe a rectangle, having given the ratio of two sides. * Ex. 1050. To divide a trapezoid into two similar trapezoids by a line parallel to the base. Ex. 1051. In the prolongation of the side AB of the triangle ABC to find a point X such that AX × BX = CX2. (322.) Ex. 1052. Through a given point P, to draw a line such that its distances from two other given points, R and S, shall have a given ratio. * Ex. 1053. Through a given point within a circle, to draw a chord so that its segments have a given ratio. * Ex. 1054. Through a point of intersection of two circles, to draw a line forming equal chords. *Ex. 1055. To construct two lines, having given their mean proportional and their difference. THEOREMS Ex. 1056. If a chord is bisected by another, either segment of the first is a mean proportional between the segments of the other. Ex. 1057. If in the triangle ABC the altitudes BD and AE meet in F, and AB = BC, then Ex. 1058. Two triangles are similar if an angle of the one is equal to an angle of the other, and the altitudes corresponding with the other angles are proportional. Ex. 1059. If between two parallel tangents, a third tangent is drawn, the radius is the mean proportional between the segments of the third tangent. Ex. 1060. If two circles are tangent externally, and through the point of contact a secant is drawn, the chords formed are proportional to the radii. Ex. 1061. If C is the mid-point of the arc AB, and a chord CD meets the chord AB in E, then of the inscribed circle is equal to the sum of the arms of the triangle. Ex. 1064. If two circles intersect, their common chord produced bisects the common tangents. Ex. 1065. If an isosceles triangle is inscribed in a circle, the tangents drawn at the vertices form another isosceles triangle. Ex. 1066. The tangents drawn at the vertices of an inscribed rectangle inclose a rhombus. Ex. 1067. Two parallelograms are similar when they have an angle of the one equal to an angle of the other, and the including sides proportional. Ex. 1068. Two rectangles are similar if two adjacent sides are proportional. Ex. 1069. Three times the sum of the squares of the sides of a triangle is equal to four times the sum of the squares of the medians. Ex. 1070. If in rectangle ABCD a perpendicular is drawn from D upon AC, the prolongation of which intersects AB in E, then AE: AD = AD: CD. Ex. 1071. Two circles are tangent externally, and through the point of contact two straight lines are drawn terminating in the circumferences; prove that the corresponding segments of the lines are proportional. Ex. 1072. If two circles intersect in A and B and the two chords BD and BC are respectively tangents to the two circles, AB is the mean proportional between AD and AC. Ex. 1073. If two circles are tangent internally, chords of the greater circle drawn from the point of contact are divided proportionally. * Ex. 1074. If in a triangle the squares of two unequal sides have the same ratio as their projections upon the third side, the triangle is a right triangle. *Ex. 1075. If from a point O, OA, OB, OC, and OD are drawn so that the angle AOB is equal to the angle BOC, and the angle BOD equal to a right angle, any line intersecting OA, OB, OC, and OD is divided harmonically. (301, 302.) * Ex. 1076. The sum of the squares of the four sides of any quadrilateral is equal to the sum of the squares of the diagonals plus four times the square of the line joining the mid-points of the diagonals. (337.) * Ex. 1077. If from a point within the triangle ABC the perpendiculars OX, OY, and OZ be drawn upon AB, BC, and CA respectively, AX2 + BY2 + CZ2 = BX2 + YC2 + ZA2. * Ex. 1078. State and prove the converse of the preceding exercise. * Ex. 1079. If two circles are tangent externally, either common external tangent is a mean proportional between the diame BOOK IV AREAS OF POLYGONS 341. DEF. The unit of surface is a square whose side is the unit of length, etc. Thus, a square 1 in. long and 1 in. wide is a square inch. 1 yd. long and 1 yd. wide is a square yard. Or a square 342. The area of a surface is the number of units of surface it contains. Thus, if the floor of a room is 25 ft. long and 15 ft. wide, it contains 15 x 25 or 375 sq. ft. Hence the area of the floor is 375 sq. ft. 343. Two figures are equivalent or equal if their areas are equal. Thus, if the area of ▲ ABC = 25 sq. ft., and the area of □ MNOP = 25 sq. ft., then ▲ ABC is equivalent to □ MNOP, or in symbols: = AABC=DMNOP. ~ NOTE. The symbol refers to the size of figures, while the symbol relates to their shape. The symbol of congruence (≈) relates to both size and shape. The symbol ~ has no meaning for figures that cannot differ in shape, e.g. for straight lines. Thence the symbol of congruence (~) when applied to straight lines has the same significance as the symbol of equality (=). 66 If the symbol of equality (=) refers to areas, it may read either "is equivalent to " or equals." Since many authors, however, designate congruent figures as equal figures, confusion may be avoided by giving preference to the term equivalent. The use of a particular symbol for equivalence (≈) cannot be recommended. PROPOSITION I. THEOREM 344. Rectangles having equal altitudes are to each other as their bases. Given R and R' the areas of two rectangles whose common altitude equals h, and whose bases are respectively b and b'. To prove Proof. CASE I. b' R b = R' b' is a rational number. Let b' n 1=m; i.e. if b' is divided into n equal parts, and one of these parts is laid off on b, b contains m such parts. If perpendiculars are erected at the points of division, R is divided into m rectangles and R' is divided into n rectangles. These rectangles are all equal. 150) b' sponding approximate value of is a rational number, it must equal the corre corresponding approximate values of and b' R are equal. |