= = 6. Given any two segments AB and CD and any two angles a and b, find a point O such that ZAOB Za and 4 COD Zb. Discuss the various possible cases and the number of points O in each case. 7. If ZAOB is a central angle of a circle and if CDE is inscribed in an arc of the same circle and if ZAOB = ≤ CDE = ÷ rt. 4, then the chords AB and CE are equal. 431. Definition. A set of lines which all pass through a common point is called a pencil of lines, and the point is called the center of the pencil. 1. The lines of a pencil intercept proportional segments on parallel transversals. Given three lines meeting in O cut by three parallel transversals. To prove that the corresponding segments are proportional, that is, to show that IB B'C' 2. If two polygons are symmetrical with respect to a point, they are congruent. (See § 170, Ex. 2.) 3. Any two figures symmetrical with respect to a point are congruent. SUGGESTION. About the point O as a pivot swing one of the figures through a straight angle. Then any point P' of the right-side figure will fall on its symmetrical point P of the left-side figure. P P 4. By means of the theorem in the preceding example show how to make an accurate copy of a map. SUGGESTION. Fasten the map to be copied on a drawing board. A long graduated ruler is made to swing freely about a fixed point 0, and by means of it construct a figure symmetrical to the map with respect to the point O. How are the distances from O measured off? 433. THEOREM. If corresponding vertices of two polygons lie on the same lines of a pencil and if they cut off proportional distances on these lines from the center, then the polygons are similar. Prove the theorem first for triangles. To prove that ▲ ABC, A'B'C', A'B'C' are similar. Prove the theorem for polygons of any number of sides. 434. Definition. Any two figures are said to have a center of similitude o, if for any two points P1 and P2 the lines P10 and P20 meet the other figure in points P'1 and P'. such that 2 P10 P20 P10 P20 Then P' and P'2 are said to correspond to the points P1 and P2. P1 Thus in the figure of § 433 O is called the center of similitude of the two polygons. Any two figures which have a center of similitude are similar. This affords a ready means of constructing a figure similar to a given figure and having some other required property that is sufficient to determine it. Definition. The ratio of any two corresponding sides of similar polygons is called their ratio of similitude. 1. Construct a polygon similar to a given polygon such that they shall have a given ratio of similitude. m SUGGESTION. Let ABCDEF be the given polygon and the n given ratio of similitude. Select any convenient point O, such that ОА = m. Draw lines OA, OB, etc. On OA lay off OA', n units. Lay off points B', C', etc., so that ОА OB OC OD OE OF = = = = OA' OB' OC' OD' OE' = Prove that A'B'C' D'E'F' is the required polygon. OF 2. Show how the preceding may be used to enlarge or reduce a map to any required size. SUGGESTION. Arrange apparatus as under Ex. 4, § 432. 3. In the figure ABCD is a parallelogram whose sides are of constant length. The point A is fixed, while the remainder of the figure is free to move. D B This shows the essential parts of an instrument called the pantograph, which is much used by engravers to transfer figures and to increase or decrease their size. The point P is made to trace out the figure which is to be copied. Hence P' traces a figure similar to it. The scale or ratio of similitude is regulated by adjusting the length of CP. 4. Construct a triangle having given two angles and the median a from one specified angle. SOLUTION. Construct any triangle ABC having the required angles and construct a median AD. Prolong DA to D', making AD' = a. Extend BA and CA and through D' draw B'C' || BC, making ▲ AB'C'. Prove that this is the required triangle. (Notice that A is the center of similitude.) 5. Inscribe a square in a given triangle using the figure given here. Compare this method with that given on page 147. 6. Construct a circle through a given point tangent to two given straight lines. b SOLUTION. Let a and b be the given lines and P the given point. Construct any circle O tangent to a and C. Draw AP meeting the circle O in C and D. Draw CO and OD and through P draw lines parallel to these meeting the bisector of the angle A formed by a and b in O' and O". Prove that O and O" are centers of the required circles. Observe that A is a center of similitude. a This method is used to construct a railway curve through a fixed point connecting two straight stretches of road. 7. On a line find a point which is equidistant from a given point and a given line. The following is another instance of the use of this very important device in constructing figures that resist other methods of attack. It consists essentially in first constructing a figure similar to the one required and then constructing one similar to this and of the proper size. 8. Given a circle with radius OE. Construct within it the design shown in the figure. That is, the inner semicircles have as diameters the sides of a regular sixteen-sided polygon. Each of the small circles is tangent to two semicircles. The outer arcs have their centers on the given circle and each is tangent to two small circles. All these arcs and circles have equal radii. SOLUTION. Construct an angle equal to one sixteenth of a perigon. Through any point D' in the bisector of this angle draw a segment EF perpendicular to the bisector and terminated by the sides of the angle. On EF as a diameter construct a semicircle. Make B'D' EF = B'A' = D'C' = = C'A' and construct the small figure. B' D E C' Now draw radii of the given circle dividing it into sixteen equal parts and bisect one of the central angles by a radius OA. Construct Z DAB = L D'A'B'. Then AB is twice the radius of the required arcs and circles. The whole figure may now be constructed. 9. Given any three non-collinear points A', B', C", to construct an equilateral triangle such that A', B', C' shall lie on the sides of the triangle, one point on each side. SUGGESTION. Through one of the points as A' draw a line such that B' and C' lie on the same side of it. 10. Given an equilateral triangle ABC, to construct a triangle similar to a given triangle A'B'C' with its vertices on the sides of ABC. SUGGESTION. Construct an equilateral triangle such that A', B', lie on its sides. Then construct a figure similar to this and of the required size. C' 11. If p and p' are similar polygons inscribed in and circumscribed about the same circle, and if 2 s is a side of the circumscribed polygon p', show that the difference of the areas of p and p' is equal to the area of a polygon similar to these and having a radius s. SUGGESTION. Let the areas of the polygons whose radii are r, r', and s be A, A', A". Prove that 12. Two circles are tangent at A. A secant through A meets the circles at B and C respectively. Prove that the tangents B at B and C are parallel to each other. 13. Prove that the segments joining one vertex of a regular polygon of n sides to the remaining vertices divide the angle at that vertex into n-2 equal parts. |