This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Math 348 (2006): Assignment #2: Solutions 1. Let ` and ` be a pair of parallel lines. Suppose the line m intersects ` and ` at points P and P , respectively. Since ` k ` , the alternate interior angles are congruent. Now let f be an isometry. We know it takes lines to lines, and preserves angles. Therefore f ( ` ), f ( ` ), and f ( m ) are all lines. Also f ( m ) intersects f ( ` ) and f ( ` ) at points f ( P ) and f ( P ), and the alternate interior angles are the same as they were before, (since f preserves angles), so they are still congruent. Therefore by the converse of the alternate interior angle theorem, f ( ` ) k f ( ` ). 2. Let r ` be reflection across a line ` . Clearly the line ` is invariant under r ` . Suppose m is another line. If m k ` , then m is not invariant, since m = r ` ( m ) will be on the opposite side of ` from m . Therefore we can suppose that m intersects ` at P , and let Q be a point on m different from P , and let R be a point on ` different from P . Then PQR = P Q R . If m is to be an invariant line, then Q must be on m also. Therefore PQR must be congruent to its supplement, so it is a right angle and m is perpendicular to ` . For a circle to be invariant under r ` , its centre O must be a fixed point, and hence must be on ` . Its easy to see that a circle centred on ` is invariant under r ` , using congruent triangles. 3. Let R O, be rotation about O by an angle . Clearly any circle centred at O is invariant under R O, , since the distance to O is preserved. Any circle centred at a point different from O cannot be invariant, since the centre would not be fixed. Consider a line ` not through O . Drop a perpendicular from O to ` at P . Then ` is tangent to the circle centred at O with radius OP . Since is less than 360 , the line ` is sent to ` , a line tangent to the same circle but now at the point P , different from P , so ` is not invariant. Its easy to see that lines though O are not invariant under R O, unless = 180 , in which case all...
View
Full
Document
This note was uploaded on 02/20/2011 for the course MATH 348 taught by Professor Karigiannis during the Summer '06 term at McGill.
 Summer '06
 Karigiannis
 Geometry, Angles

Click to edit the document details