5.1+Perpendiculars+&+Bisectors


 * 5.1 Perpendiculars & Bisectors** (Pages 264-271) **Dylan Humphrey**

The topic of this section is to use perpendiculars and bisectors. The goal of this section is to use properties of angle bisectors to identify equal distances. Section 5.1 says that perpendiculars bisect a segment. Now, the end points on the segment will be equidistant from each other. Also, if a point is on the perpendicular bisector it is equidistant from the two endpoints of the segment. You will learn that if a point is on an angle bisector it is equidistant from the two sides of the angle. Lastly, if a point is in the interior of an angle and is equidistant from the sides of the angle, then it lies on the bisector of the angle.




 * Construct a Perpendicular Bisector**

A **perpendicular bisector** of a line segment //AB// is a line that divides the line //AB// into two equal parts at a right angle.

// **Example** // **//://** Construct a perpendicular bisector of the given line segment //AB//.
 * //Solution://**
 * Step 1** : Stretch your compasses until it is more then half the length of //AB//. Put the sharp end at //A// and mark an arc above and another arc below line segment //AB//.


 * Step 2 ** : Without changing the width of the compasses, put the sharp end at //B// and mark arcs above and below the line segment //AB// that will intersect with the arcs drawn in step 1.
 * Step 3 ** : Join the two points where the arcs intersect with a straight line. This line is the perpendicular bisector of //AB//. //P// is the midpoint of //AB//.

" http://www.onlinemathlearning.com/perpendicular-bisector.html "

A. 6 cm and 8 cm B. 7 cm and 7 cm C. 6 cm and 6 cm D. 7 cm and 6 cm
 * Example 1:** AB is a line segment of length 12 cm. CD is the perpendicular bisector of AB. Find the length of AO and OB.
 * Choices:**


 * Correct Answer: C**


 * Step 1:** A perpendicular bisector divides a line segment into two equal parts.
 * Step 2:** The length of AB is 12 cm and CD is the perpendicular bisector of AB.
 * Step 3:** So, AO = OB = [[image:http://www.icoachmath.com/Sitemap/images/Perpendicular%20Bisector3.gif width="29" height="41"]] = [[image:http://www.icoachmath.com/Sitemap/images/Perpendicular%20Bisector4.gif width="21" height="41"]] = 6 [Substitute AB = 12.]
 * Step 4:** AO = OB = 6 cm


 * Example 2:** What is the length of AB, if line //l// is the segment bisector and AO = 6 units?

A. 13 units B. 6 units C. 12 units D. 14 units
 * Choices:**


 * Correct Answer: C**

[As the segment bisector passes through the midpoint of the segment.]
 * Step 1:** Line //l// divides AB into two equal parts and O is the midpoint of AB.
 * Step 2:** AB = 2(AO) = 2(6) = 12 [Substitute AO = 6.]
 * Step 3:** So, the length of AB is 12 units.

" http://www.icoachmath.com/Sitemap/PerpendicularBisector.html "


 * Example 3:** If BC is the perpendicular bisector of AD and AE=8, what is ED?


 * Solution:** AD=8 because of the segment bisector.

(This example was made in sketchpad.)


 * Vocabulary**

Perpendicular Bisector- a segment, ray or line that intersects a segment at its midpoint. Perpendicular bisectors create a right angle at the point of intersection.

Equidistant from two points- a point is equidistant from two points if its distance from each point is the same.

Distance from a point to a line- the distance from a point to a line is defined as the length of the perpendicular segment from the point to the line.

Equidistant from the two lines- when a point is the same distance from one line as it is form another line.


 * THEOREM 5.1 Perpendicular Bisector Theorem**

If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.If is the perpendicular bisector of, then //CA// = //CB//.


 * THEOREM 5.2 Converse of the Perpendicular Bisector Theorem**

If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.If //DA// = //DB//, then //D// lies on the perpendicular bisector of.


 * THEREOM 5.3 Angle Bisector Thereom**

If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle.

If //m////BAD// = //m////CAD//, then //DB// = //DC//.


 * THEREOM 5.4 Converse of the Angle Bisector Thereom**

If a point is in the interior of an angle and is equidistant from the sides of the angle, then it lies on the bisector of the angle.

If //DB// = //DC//, then //m////BAD// = //m////CAD//.

" http://delta.classwell.com/ebooks/navigateBook.clg?sectionType=unit&navigation=1&prevNext=1&curSeq=260&curDispPage=264&xpqData=%2Fcontent " Online Textbook. (Pages 265-266)




 * Practice Worksheets** (with answers)