EXERCISES.

85. Given the middle point of a chord of a circle,

to construct the chord.

(To draw through C a chord which is bisected at C.)

86. To draw a line tangent to a given circle and parallel to a given straight line.

(To draw a tangent || AB.)

87. To draw a line tangent to a given circle and perpendicular to a given straight line.

88. To draw a straight line through a given point within a given acute 2, forming with the sides of Athe angle an isosceles triangle.

89. Given the base, an adjacent angle, and the altitude of a triangle, to construct the triangle.

(Draw a || to the base at a distance equal to the altitude.)

90. Given the base, an adjacent side, and the altitude of a triangle, to construct the triangle.

Discuss the problem for the following cases: 3. n <p.

1. n> p. 2. n = p.

91. To construct a rhombus, having given its base and altitude. (Draw a to the base at a distance equal to the altitude.) What restriction is there on the values of the given lines?

92. Given the altitude and the sides including the vertical angle of a triangle, to construct the triangle.

What restriction is there on the values of the given lines?

Discuss the problem for the following cases: 1. m <n or> n.

2. m = n.

93. Given the altitude of a triangle, and the angles at the extremities of the base, to construct the triangle.

(The between the altitude and an adjacent side is the complement of the at the extremity of the base, if acute, or of its supplement, if obtuse.)

94. To construct an isosceles triangle, having given the base and the radius of the circumscribed circle.

What restriction is there on the values of the given lines ?

95. To construct a square, having given one of its diagonals. (§ 195.)

96. To construct a right triangle, having given the hypotenuse and the length of the perpendicular drawn to it from the vertex of the right angle.

What restriction is there on the values of the given lines?

D

n

B

m

97. To construct a right triangle, having given the hypotenuse and

a leg.

What restriction is there on the values of the given lines?

98. Given the base of a triangle and the

99. To describe a circle of given radius tangent to two given intersecting lines.

(Draw a || to one of the lines at a distance equal to the radius.) 100. To describe a circle tangent to a given straight line, having its centre at a given point without the line.

101. To construct a circle having its centre in a given line, and passing through two given points without the line. (§ 163.)

What restriction is there on the positions of the given points?

102. In a given straight line to find a point equally distant from two given intersecting lines. (§ 101.)

103. Given a side and the diagonals of a parallelogram, to construct the parallelogram.

What restriction is there on the values of the given lines?

104. Through a given point without a given straight line, to describe a circle tangent to the given line at a given point. (§ 163.)

E

105. Through a given point within a circle to draw a chord equal to a given chord. (§ 164.) What restriction is there on the position of the given point?

106. Through a given point to describe a circle of given radius tangent to a given straight line.

(Draw a || to the given line at a distance equal to the radius.)

107. To describe a circle of given radius tangent to two given circles.

(To describe a ○ of radius m tangent to two given whose radii are n and p, respectively.)

What restriction is there on the value of m?

108. To describe a circle tangent to two given parallels, and passing through a given point.

What restriction is there on the position of the given point?

109. To describe a circle of given radius, tangent to a given line and a given circle.

(Draw a || to the given line at a distance equal to the given radius.) 110. To construct a parallelogram, having given a side, an angle, and the diagonal drawn from the vertex of the angle.

111. In a given triangle to inscribe a rhombus, having one of its angles coincident with an angle of the triangle.

(Bisect the which is common to the ▲ and the rhombus.)

112. To describe a circle touching two given intersecting lines, one of them at a given point. (§ 169.)

113. In a given sector to inscribe a circle.

(The problem is the same as inscribing a in ▲ O'CD.)

-D

B

114. In a given right triangle to inscribe a square, having one of its angles coincident with the right angle of the triangle.

115. Through a vertex of a triangle to draw a straight line equally distant from the other vertices.

116. Given the base, the altitude, and the vertical angle of a triangle, to construct the triangle. (§ 226.)

(Construct on the given base as a chord a segment which shall contain the given 2.)

117. Given the base of a triangle, its vertical angle, and the median drawn to the base, to construct the triangle.

118. To construct a triangle, having given the middle points of its sides.

119. Given two sides of a triangle, and the median drawn to the third side, to construct the triangle.

(Construct▲ ABD with its sides equal to

m, n, and 2p, respectively.)

What restriction is there on the values of

the given lines?

120. Given the base, the altitude, and the radius of the circumscribed circle of a triangle, to construct the triangle.

(The centre of the circumscribed O lies at a distance from each vertex equal to the radius of the O.)

121. To draw common tangents to two given circles which do not intersect.

(To draw exterior common tangents, describe OAA' with its radius equal to the difference of the radii of the given .

To draw interior common tangents, describe OAA' with its radius equal to the sum of the radii of the given ©.)

Note. For additional exercises on Book II., see p. 224.

BOOK III.

THEORY OF PROPORTION.-SIMILAR

POLYGONS.

DEFINITIONS.

227. A Proportion is a statement that two ratios are equal.

228. The statement that the ratio of a to b is equal to the ratio of c to d, may be written in either of the forms

229. The first and fourth terms of a proportion are called the extremes, and the second and third terms the means.

The first and third terms are called the antecedents, and the second and fourth terms the consequents.

Thus, in the proportion a: bc:d, a and d are the extremes, b and c the means, a and c the antecedents, and b and d the consequents.

230. If the means of a proportion are equal, either mean is called a mean proportional between the first and last terms, and the last term is called a third proportional to the first and second terms.

Thus, in the proportion a: b = b: c, b is a mean proportional between a and c, and c a third proportional to a and b.

231. A fourth proportional to three quantities is the fourth term of a proportion, whose first three terms are the three quantities taken in their order.

Thus, in the proportion a: b = c: d, d is a fourth proportional to a, b, and c.