SUG. 4. Compare O M with O F, O F with O P, and, finally, O M with O P.

SUG. 5. Compare S N with O P.

Therefore

PROPOSITION XII. THEOREM.

154. A straight line perpendicular to the radius of a circle at its extremity is tangent to the circle.

MB

Let O B represent a straight line perpendicular to the radius O S at its extremity O.

To prove that O B is tangent to the circle.

SUG. 1. The point O is common to the circle S and the line OB. Why?

SUG. 2. Let M represent any other point in O B Draw S M.

SUG. 3. Compare S O and S M in respect to length. SUG. 4. Where, then, is M with respect to the circle ? See Art. 139.

Therefore

PROPOSITION XIII. THEOREM.

155. Converse of Prop. XII If a straight line is tangent to a circle the radius meeting it at the point of contact is perpendicular to it.

MB

S

Let O B represent a tangent to the circle S, O the point of contact, and S O the radius drawn to the point of contact.

To prove that S O is perpendicular to O B.

SUG. 1. Draw S M, any other line from S to O B. SUG 2. Compare S O and S M in respect to length. SUG. 3. See article 108.

Therefore

156. COROLLARY. At any point in a circumference, one, and only one, tangent can be drawn and a straight line perpendicular to a tangent at the point of contact passes through the center.

PROPOSITION XIV. THEOREM.

157. Arcs of a circle intercepted by parallel chords are equal.

M

B

Let A B and C D represent two parallel lines intercepting the arcs A C and B D.

To prove that arċ A C equals arc B D.

SUG. 1. Drop a

from 0 to CD, and extend it to

meet the circumference at M.

SUG. 2. How is O M related to A B? See Art. 85. SUG. 3. See Prop. IX, and complete the demonstration.

Therefore

Ex. 101. If one of the equal sides of an isosceles triangle be extended through the vertex and the exterior angle formed be bisected, prove that the bisector is parallel to the base.

Ex. 102. If the diagonals of a parallelogram are equal, prove that the parallelogram is a rectangle.

Ex. 103. The bisectors of the interior augles of a parallelogram form a rectangle.

PROPOSITION XV. THEOREM.

158. Through three points, not in the same straight line, one circumference, and only one, is possible.

Let A, B and C represent three given points not in the same straight line.

To prove that through A, B and C one circumference, and but one, is possible.

SUG. 1. What is the locus of a point equally distant from A and C? Give auth.

SUG. 2. What is the locus of a point equally distant from A and B? Give auth.

SUG. 3. Will these two loci intersect? Why?

SUG. 4. Is there a point equally distant from A, B, and C?

SG. 5. Can a circumference pass through the points A, B and C?

SUG. 6. Can there be more than one such circumference? Why?

Therefore

MEASUREMENT.

159. Anything which can be measured by a unit of the same kind is called quantity.

The quantities used in geometry are the geometric magnitudes, viz., lines, surfaces and solids.

To measure a quantity is to find out how many times it contains another selected quantity of the same kind, called the unit of measure.

In every day experience, the unit of measure is a standard accepted by general consent; as a foot, a square yard, a ton, a cord, etc.

160. The measure of a quantity is the number which expresses how many times the unit of measure is contained in the given quantity.

A careful distinction should be made between number, or the measure of quantity, and quantity. These words are sometimes carelessly confused. Twenty-six books, 3 ft., 5 pints and 29 sq. ft. are quantities; 26, 3, 5 and 29 are numbers. The units by which we derive these numbers from the quantities are, respectively, a book, a foot, a pint and a square foot.

161. The ratio of two quantities is the number of times the first contains the second; i. e., the measure of the first regarding the second as the unit of measure; or, having measured both quantities by the same unit, the quotient of the measure of the first divided by the measure of the second.

The ratio of line A to line B is the number of times A contains B, which may be determined by laying off B upon A. Or, if A and B be measured by same unit m, the ratio of A to B is the number of times the

A

B

m

measure of A contains the measure of B. Suppose m