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The tree rooted by the root container is known as the range s context tree When navigating XML documents, the traversal-range recommendation de nes two separate interfaces for XML elements The NodeIterator interface provides methods to move linearly through a document, traversing it from node to node in the order in which they occur The TreeWalker interface permits traversal of the document as a tree-based structure To use either of these interfaces, call the create method, and set appropriate ags for elements that should be included in the view In practice, we would use an application such as Xerces (available from the Apache Software Foundation), which is a Java application that has all the les you need and plenty of documentation Others are available, of course, from Microsoft, Sun, and other vendors To traverse a at representation of XML, use the NodeIterator interface.



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11.5 Proving a locus theorem Prove that the locus of points equidistant from two given points is the perpendicular bisector of the segment joining the two points.

First prove that any point on the locus satisfies the condition: Given: Points A and B. CD is the ' bisector of AB. To Prove: Any point P on CD is equidistant from A and B; that is, PA > PB. Plan: Prove ^PEA > ^PEB to obtain PA > PB. PROOF:

Statements 1. CD is the ' bisector of AB. 2. /PEA > /PEB 3. 4. 5. 6. AE > EB PE > PE ^PEA > ^PEB PA > PB

Reasons 1. Given 2. Perpendiculars form right angles; all right angles are congruent. 3. To bisect is to divide into congruent parts. 4. Reflexive property 5. SAS 6. Corresponding parts of > triangles are >.





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Then prove that any point satisfying the condition is on the locus: Given: Any point Q which is equidistant from points A and B (QA > QB). To Prove: Q is on the perpendicular bisector of AB. Plan: Draw QG perpendicular to AB and prove by congruent triangles that QG bisects AB. PROOF:

Statements 1. Draw QG ' AB . 2. QA > QB 3. /QGA and /QGB are rt. /s; ^QGA and ^QGB are rt. ^s. 4. QG > QG 5. ^QGA > ^QGB 6. AG > GB 7. QG bisects AB . 8. QG is ' bisector of AB.

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Reasons 1. Through an external point, a line can be drawn perpendicular to a given line. 2. Given 3. Perpendiculars form right angles; ^s with a rt./ are rt. ^s. 4. Reflexive property 5. Hy-leg 6. Corresponding parts of > triangles are >. 7. To bisect is to divide into two congruent parts 8. A line perpendicular to a segment and bisecting it is its perpendicular bisector.

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11.1. Determine the locus of (a) The midpoints of the radii of a given circle (b) The midpoints of chords of a given circle parallel to a given line (c) The midpoints of chords of fixed length in a given circle (d) The vertex of the right angle of a triangle having a given hypotenuse (e) The vertex of an isosceles triangle having a given base (f) The center of a circle which passes through two given points (g) The center of a circle tangent to a given line at a given point on that line (h) The center of a circle tangent to the sides of a given angle 11.2. Determine the locus of (a) A boat moving so that it is equidistant from the parallel banks of a stream (b) A swimmer maintaining the same distance from two floats (c) A police helicopter in pursuit of a car which has just passed the junction of two straight roads and which may be on either one of them (d) A treasure buried at the same distance from two intersecting straight roads 11.3. Determine the locus of (a) a planet moving at a fixed distance from its sun; (b) a boat moving at a fixed distance from the coast of a circular island; (c) plants laid at a distance of 20 ft from a straight row of other plants; (d) the outer extremity of a clock hand. (11.1) (11.1) (11.1)

11.4. Excluding points lying outside rectangle ABCD in Fig. 11-15, find the locus of points which are (a) Equidistant from AD and BC (b) Equidistant from AB and CD (c) Equidistant from A and B (d) Equidistant from B and C (e) 5 units from BC (f) 10 units from AB (g) 20 units from CD (h) 10 units from B

(11.1)

Fig. 11-15

Fig. 11-16

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