72. Draw a rectangle having one side 2 inches in length, and subtending an ▲ of 40° at the point of intersection of the diagonals.

(Use a protractor in Exercises 72 to 86.)

73. Draw a gm with diagonals 2 inches and 4 inches and their of intersection 50°.

74. Draw a gm with diagonals 4 inches and 7 inches and one side 5 inches.

75. Draw a ||gm with side 3 inches, diagonal 2 inches and 35°. Show that there are two solutions.

76. Draw a gm with side 23 inches, 70° and diagonal opposite of 70° equal to 4 inches.

77. Draw a rectangle having the perimeter 8 inches and an between the diagonals 80°.

78. Draw a rectangle having the difference of two sides 1 inch and an between the diagonals 50°.

79. Draw a rectangle which has the perimeter 9 inches and a diagonal 3 inches.

80. Draw an ▲ of 55°. Find within the a point which is 1 inch from one arm and 2 inches from the other.

81. Construct a ▲ in which side a 10.6 cm. and A

=

78°.

82. Construct a ▲ with perimeter 4 inches and s 70° and 50°.

83. AB, CD are two || st. lines; P, Q two fixed points. Find a point equidistant from AB, CD and also equidistant from P and Q. When is this impossible?

84. Through two given points on the same side of a given st. line draw two st. lines so as to form with the given st. line an equilateral A.

85. Construct a rhombus with one diagonal 2 inches and the opposite

100°.

86. Construct a ▲ in which a = 8 cm., b LC = 50°.

87. Squares ABGE, ACHF are described externally on two sides of a ABC Prove that the median AD of the EF and equal to half of EF.

A is

NOTE.-Rotate ▲ ABC through a

coincide with AF.

rt. L making AC

88. Prove also in Ex. 87 that EC is 1 and

89. Trisect a rt. L.

= BF.

90. From any point in the base of an isosceles

st.

lines are drawn || to the equal sides and terminated by them. Prove that the sum of these lines = one of the equal sides.

91. ABC is a st. line such that AB = BC. Is are drawn from A, B, C to another st. line EF. Prove that the from B half the sum of the 1s from A and C, unless EF passes between A and C, and then the from B = half the difference of the Is from A and C.

=

92. AD is the bisector of LA of AABC, and M the middle point of BC. BE and CF are AD. Prove that

ME = MF.

93. E, F are the middle points of AD, BC respectively in the gm ABCD. Prove that BE, DF trisect AC.

94. Find a point P in the side AC of a ▲ ABC so that AP may be equal to the from P to BC.

95. If the st. line AB be bisected at C and produced to D, prove that CD is half the sum of AD, BD.

96. In ABC side AC > side AB; AX 1 BC and AD is a median. Prove that (1) < CAX > < BAX; (2) / CAD < BAC falls between AX and

AD.

DAB; (3) the bisector of

97. The median of a ▲ ABC drawn from A is not less than the bisector of LA.

98. In a quadrilateral ABCD, AB = DC and L B Prove that AD || BC.

= L C.

99. If two medians of a ▲ are equal, the ▲ is isosceles.

NOTE.-Use Ex. 6, § 64.

100. If both pairs of opposites of a quadrilateral are equal, the quadrilateral is a gm.

101. Find the point on the base of a ▲ such that the difference of the 1s from it to the sides is equal to a given st. line.

102. Find the point on the base of a ▲ such that the sum of the Is from it to the sides is equal to a given st. line.

103. Show that the three exterior Ls at A, C, E, in the hexagon ABCDEF, are together less than the three interior s at B, D, F by two rt. s.

AREAS OF PARALLELOGRAMS AND TRIANGLES

70. A square unit of area is a square, each side of which is equal to a unit of length.

Examples:-A square inch is a square each side of which is one inch; a square centimetre is a square each side of which is one centimetre.

The acre is an exceptional case.

71. A numerical measure of any area is the number of times the area contains some unit of area.

ABCD is a rectangle one centimetre wide and five centimetres long.

A

B

This rectangle is a strip divided into five square centimetres, and consequently the numerical measure of its area in square centimetres is 5.

72. ABCD is a rectangle 3 cm. wide and 5 cm. long.

This rectangle is divided into 5 strips of 3 sq. cm. each, or into 3 strips of 5 sq. cm. each, and consequently

the measure of the area in square centimetres is 5 × 3 sq. cm., or 3 x 5 sq. cm.

Similarly, if the length of a rectangle is 2:34 inches and its breadth 56 of an inch, the one-hundreth of an inch may be taken as the unit and the rectangle can be divided into 234 strips each containing 56 square one-hundreths of an inch. The measure of the area then is 234 × 56 of these small squares, ten thousand (100 x 100) of which make one square inch.

This method of expressing the area of a rectangle may be carried to any degree of approximation, so that in all cases the numerical measure of its area is equal to the product of its length by its breadth. In a rectangle any side may be called the base, and then either of the adjacent sides is the altitude.

A rectangle, as ABCD, is commonly represented by the symbol AB. BC, where AB and BC may be taken to represent the number of units in the length and the breadth respectively.

Or, if a be the measure of the base of a rectangle and b the measure of its altitude, the area is ab.

In the case of a square, the base is equal to the altitude, and if the measure of each be a, the area is a2.