40, 60 and 65 millimetres. What relation does the angle opposite the greatest side bear to a right angle? 15. Construct a triangle with sides of 2, 3 and 3 inches (3.52<22+32). What is the angle opposite the side of 3 inches? 16. Construct any quadrilateral with its diagonals at right angles to each other. Show that the sum of the squares on two opposite sides is equal to the sum of the squares on the other two sides. 17. Describe a square ABCD, and in the sides take points E, F, G, H, such that AE=BF=CG=DH. What is the figure EFGH. Apply tests. Give reasons. 18. Two squares being given, say of 9 and 16 square inches, show how to draw a line the square on which shall be equal to the difference of these given squares. 19. ABC, A'B'C' are right-angled triangles with the hypotenuses AB, A'B' equal, and also the sides BC, B'C' equal. the remaining sides AC, A'C' are equal. 20. The sides of a triangle are 1, 2, 2 inches. square equal to it. 21. The side of an equilateral triangle is 2 inches. square equal to it. Show that Construct a Construct a 22. The sides of a rectangle are 24 and 54 sixteenths of an inch. Construct a square equal to it. Measure the side of the square, and thence verify the accuracy of your construction. 23. If a right-angled triangle have one of the acute angles double the other, divide it into two triangles, one equilateral and the other isosceles. 24. Bisect the hypotenuse of a right-angled triangle. What relation between the distances of the point so obtained from the three angles? 25. ABC is a right-angled triangle, and CD is drawn perpendicular to the hypotenuse. Examine the relations between the angles of the three triangles ABC, ACD, BCD. Give reasons. The Circle. Its Symmetry. Tangents. Finding of Centre. 1. The fundamental quality of the circle, next to the equality of its radii, is its symmetry. In the first place, every line drawn through the centre from circumference to circumference (i.e., every diameter) is bisected at the centre. This is called central symmetry. In the second place, every chord drawn at right angles to a diameter is bisected by that diameter. This is called axial symmetry. Thus the chord EFG being perpendicular to OA, the parts EF, FG are equal. Measurement will establish the equality of these parts. Or we may prove it thus: E The rt. les at F are equal. Also sides EO, OF= sides GO, OF. .. (Ch. III., 2) EF=FG. And hence all chords perpendicular to a diameter are bisected by it. 2. As the chord BCD moves parallel to itself down to A, since the parts on each side of the diameter are always equal, when one part vanishes, the other vanishes also. Thus the line TAP, through A parallel to BCD, while it touches the circle, does not cut it. Such a line (TAP) is called the tangent to the circle at A. That is to say, a tangent is a line drawn through the extremity of a diameter, and at right angles to it. The tangent is evidently a straight line which meets. the circle, but does not cut it: this is sometimes given as the definition of a tangent. 3. Since a diameter bisects every chord to which it is at right angles, therefore a line drawn through the bisection of a chord and at right angles to it, must be a diameter. Hence if the centre of any circle be not indicated, we may reach it by the following construction: Draw any chord AB. Bisect it at C. Draw DCE perpendicular to AB. DE must pass through the centre. Hence, bisecting DE at F, F must be the centre of the circle. We may describe circles without marking their centres by placing a piece of thin wood or cardboard under the station A ary point of the compasses, removing this piece of wood or cardboard when the circle is described. Circles being thus described, or being obtained by marking with the pencil about a round object placed on the paper (coin, bottom of ink bottle, plate, &c.), attempts should be made to locate the centre by the eye's judgment. We may afterwards test the correctness of this by making the preceding construction, and finally test the accuracy of the construction by trying with such centre to reproduce the circle by using the compasses. It will be found, of course, that the greater the circle, the greater will be the difficulty of locating, with the eye's judgment, the position of the centre. The same difficulty occurs in locating the bisection of a straight line with the eye. 4. If only an arc of the circle be given, we may find the centre, and complete the circle, as follows: B Draw two chords AB and CD; find their middle points E and F; through these middle points draw perpendiculars EG and FH. The centre of the circle must lie on each of the lines EG and FH (Ch. X., 3), and therefore must be at 0. Ares of circles should be described without marking the centres, by the method suggested in § 3. The positions of the centres should then be judged with the eye; afterwards constructed for, and the accuracy of the construction tested by attempting, with the compasses, to describe the arc with the centre so obtained. 5. If any line AB be bisected at C, and CD be drawn perpendicular to it, then all points in CD are equally distant from A and B. Hence if we place the sharp point of the compasses at any point on CD, and the pencil end at A, and describe a circle, it will also pass through B. We thus get an unlimited number of circles through A and B, all of which have their centres at different points on CD. Draw a line AB of 50 millimetres, and describe circles passing through A and B, with radii 30, 40, 50 and 60 millimetres. 6. We can readily obtain a method for describing a circle to pass through any three points: Let A, B, C be the three points. Draw DO from the middle point of AB at right angles to it; and draw EO from the middle point of BC at right angles to it. Then all points in DO are equally distant from A and B; and all points in EO are equally distant from B and C. Hence O is equally distant from A, B and C; and placing the sharp point of the compasses at O and the pencil end at A, and describing a circle, it will pass through B and C, if the construction has been accurate. AB is 1 inch, BC is 2 inches, and angle ABC is 120°. Describe a circle to pass through A, B and C. AB is 40 and BC 60 millimetres, and the angle ABC is 75°. Describe a circle to pass through A, B and C. AB is 1 and BC 2 inches, and the angle ABC is 90°. Describe a circle to pass through A, B and C. Show that its centre bisects AC. Mark sets of three points in various positions with respect to one another, and describe a circle to pass through each set. |