QUESTIONS AND EXERCISES IN THE PRECEDING PROPOSITIONS. A. (1) In describing an equilateral triangle (23) upon a given straight line, how much is taken for granted? If a second triangle be drawn on the opposite side of the line by a similar construction, what figure will the two together make? (2) Have we yet laid down any mode of measuring an angle? If not, how are we able to prove that one angle is equal to, less than, or greater than, another according to circumstances? (3) If the angles of one triangle be equal to the angles of another, each to each, are the triangles necessarily equal? What is the force of the expression, 'each to each'? Exhibit a case where the angles are respectively equal, but not 'each to each'. (4) What is meant in (25) by a 'given angle'? Is it necessary that the triangle DEF should be equilateral? What other triangle would do as well? Does it matter on which side of DE the triangle is described? (5) Define an 'isosceles' triangle. Is an equilateral triangle isosceles? Can more than one isosceles triangle be constructed on the same base and on the same side of it? Can a right-angled triangle be also isosceles ? (6) What is the precise meaning of 'given straight line' in (27), where it is required to bisect it? Is it the same as in (28) and (29)? If not, what is the dif ference? (7) What is the meaning of the word 'base' as applied to a triangle, and to a parallelogram? Is it restricted to one fixed side only? (8) Shew that only one straight line can be drawn perpendicular to a given straight line from a given point without it. (9) Shew that the perpendicular is the shortest of all lines from a given point to a given straight line. Of all such lines which measures the distance of the point from the given line? (10) If in (30) straight lines be drawn bisecting each of the angles ACD, BCD, shew that these straight lines are at right angles to one another. (11) Shew that any point in the straight line bisecting an angle is equidistant from the two straight lines forming the angle. (12) Shew that any side of a triangle is less than half the sum of all three sides of the same triangle. (13) Shew that the straight line drawn from the middle point of the base of an isosceles triangle to the vertex of the opposite angle is at right angles to the base, and bisects the opposite angle. (14) Shew that each angle of an equilateral triangle is two-thirds of a right angle. Trisect a right angle. (15) Can a triangle have more than one of its angles a right angle, or an obtuse angle? If not, why not? (16) Shew that the four angles of every quadrilateral figure are together equal to four right angles. (17) If one of the angles of a parallelogram be a right angle, does this determine all the other angles? (18) Shew that any two straight lines at right angles to the same straight line, and on the same side of it, are parallel. (19) If two parallel straight lines be intersected by two other parallel straight lines, shew that the parts of the latter two intercepted between the former two are equal to each other. (20) If two straight lines in the same plane be equal and parallel, shew that the straight lines joining their extremities towards the same parts are also equal and parallel. (21) If two straight lines be drawn bisecting two angles of a triangle, shew that the point in which they intersect is equidistant from the three sides of the triangle. (22) Is it correct to speak of drawing a line from an angle? The expression is found in Simson's Euclid; what does it mean? See definition of angle. (23) Simson also, after defining 'vertex' of an angle, on the first occasion of using the term (Prop. vII) speaks of the 'vertex' of a triangle. What is the difference betwixt the two? (24) Shew that the straight line drawn from the vertex of the right angle in a right-angled triangle to the middle point of the hypothenuse is equal to half the hypothenuse. (25) Explain what is meant by the square of a line. Is the square of the line AB the same as the square of the line BA? (26) Take the particular case of a right-angled triangle which is isosceles, and shew how the squares described on the two sides can be made to cover the square on the hypothenuse. square (27) Is the of AB double of the square of half AB? If not, what then? (28) Make a square which shall be double of the square of a given line. (29) Is the rectangle contained by AB, BC, the same as that contained by CB, BA? or that contained by BC, AB? (30) Make a rectangle which shall be double of a given rectangle AB, BC. (31) Is it certain à priori that either a square, or an equilateral triangle, according to Definition, is possible? Explain fully. (32) Make a right-angled triangle which shall be double of a given right-angled triangle. (33) Can a triangle be equal to a rectangle? If so, draw a rectangle equal to a given triangle. (34) Each of the sides of a rectangle is double of the corresponding side of another rectangle; how many times does the larger rectangle contain the other? (35) If a side of an equilateral triangle be double of the side of another equilateral triangle, what proportion will the two triangles bear to each other? (36) Shew that the diagonals of a square bisect each other at right angles. (37) Shew that in every parallelogram the squares of the diagonals are together equal to the sum of the squares of all the sides. THE CIRCLE AND STRAIGHT LINES CONNECTED WITH IT. 48. DEFINITIONS. An ARC of a circle is a portion of the circumference of the circle. A CHORD is the straight line which joins the two extremities of an arc. A segment of a circle is a portion of a circle bounded by an arc and its chord. A sector of a circle is a portion of a circle bounded by an arc and two radii drawn to the two extremities of the arc. B Thus, in the annexed fig. the curved line from A to B is an arc, the straight line AB is a chord, the area enclosed between the arc and the chord AB is a segment, and the area enclosed between the arc and the two radii OA, OB is a sector, of the circle ABD whose centre is 0. Hence a diameter is a particular chord; a semi-circle is a particular segment; and a quadrant is a particular sector. The learner must keep in mind the difference between arc, segment, and sector. Observe, that an arc is a line; but a segment, and a sector, are both areas. Observe also that, according to the Definition, the straight line AB is the chord of the arc ADB as well as of the arc AB; but the smaller arc of the two is always meant except when it is otherwise expressed. 49. PROP. I. A straight line drawn from the centre of a circle to the middle point of a chord is perpendicular to that chord. Let AB be the chord of any arc AB of a circle whose centre is O*; and C the middle point in the chord. Join OC; then OC shall be perpendicular to the chord AB. It is not necessary here to determine the precise position of the centre, but merely to assume, according to the definition, that there is such a point somewhere within the circle, and to call it the point. O. For, joining OA, OB, in the two triangles OAC, OBC, the two sides OA, AC, are equal to the two sides OB, BC, each to each; and ▲ OAC = = < OBC, since OA = OB (26), .. the triangles are equal in all respects (24), and < OCA=¿ ÒCB, and.. each of them is a right angle; that is, OC is perpendicular to AB. .. B COR. Conversely, if a straight line be drawn from the middle point of a chord at right angles to the chord, that straight line shall pass through the centre of the circle. Also, a perpendicular drawn from the centre of a circle to a chord will bisect the chord. 50. PROP. II. To find the centre of a given circle. Let the annexed fig. be a given circle; and let it be required to find its centre. Take any two points A, B, in the circumference, and join AB; bisect AB in C; and from C draw CD at right angles to AB. Then the centre of the circle is somewhere in the line CD (49). Again take two other points E, F in the circumference; join EF; bisect EF in G; BE and draw GH at right angles to EF, intersecting CD in the point O. Then the centre of the circle is in GH; and it is also in CD; but CD and GH have only one point in common, viz. the point 0; .. O is the centre of the circle. COR. The same method evidently applies to the case of a given segment, or arc, when the centre of the circle to which it belongs is required, or when it is required to complete the circle. Another Method. Draw the chord AB; bisect it in By a given circle is here meant a plane surface presented to us in the form of a circle, as a crown-piece, or the end of a round ruler. Or it is a circle whose circumference is traced out upon a plane surface. |