Aim #9: How do we construct a perpendicular bisector? CC Geometry H Do Now: Using the angle below: 1. Bisect the angle. Label the bisector BD. 2. Construct a copy of ≮DBC using vertex B'.

A

B C B'

Relevant Vocabulary Two lines are PERPENDICULAR ( ) if they intersect in one point, and any of the angles formed by the intersection of the lines is a ______angle. (Two segments or rays are perpendicular if the lines containing them are .) The MIDPOINT of a segment divides a segment into 2 = or _____ parts. A SEGMENT BISECTOR passes through the ______of a segment. An ANGLE BISECTOR is a ray (line/segment) that divides an _____ into 2 = or ≅ parts. AJ has been constructed as the angle bisector of ≮BAD. B • Draw BD. • Label C, the point of intersection of BD and AJ. A J Notice: (1) BC = CD so C is a ______. D Therefore AJ is a ______of BD.

(2) ≮BCA = ≮DCA. Each of these angles measures____. Therefore, BD and AJ are ______.

AJ is the ______of BD.

The perpendicular bisector of a line segment passes through the ______of the segment and forms ______angles with the segment. (It is perpendicular to a segment at its midpoint.) Using a compass and straightedge, we will now construct a perpendicular bisector of a line segment. Experiment with your construction tools and the following line segment to establish the steps that result in the perpendicular bisector. [Use what you know about constructing an equilateral triangle.] http://www.mathsisfun.com/geometry/construct­linebisect.html

A B

Steps for constructing a perpendicular bisector:

1. Draw circle A: center A, radius > 1/2AB and circle B: center B, radius > 1/2BA. 2. Label the points of intersections as C and D. 3. Draw CD.

Construct the perpendicular bisector of CD. • Label the midpoint M and label the perpendicular bisector as EF.

• Name one right angle:______.

C

D Relevant Vocabulary: EQUIDISTANT A point A is said to be equidistant from two different points B and C if AB = AC. Draw a diagram.

CDE is the perpendicular bisector of AB. (C, D, and E are collinear points.)

Using your compass, what conclusion can you make about the following pairs of segments? 1) AC and BC equal, circle C with rad. CA goes through pt B. 2) AD and BD equal, circle D with rad. DA goes through pt B. 3) AE and BE equal, circle E with rad. EA goes through pt B. Based on your findings, fill in the observation below. Any point on the perpendicular bisector of a line segment is ______from the endpoints of the line segment.

Why? Any circle we constructed w/a center on the perp. bisector and a radius that passed through one endpt, also passed through the other endpt. All radii of a circle are equal, so these segments must be equal.

Now construct a perpendicular line to line l from a point X not on line l . The steps of the construction have been outlined below for you.

X l

Step 1: Draw circle X so that the circle intersects line l in two points. Step 2: Label the two points of intersection as B and C. Step 3: Draw circle B: center ____, radius ______. Step 4: Draw circle C: center ____, radius ______. Step 5: Label the unlabeled intersection of circle B and circle C as D. Step 6: Draw the perpendicular bisector: line ______. Exercises 1. Divide segment AB into 4 segments of equal length.

A B

2. Construct parallel lines l1 and l2 as follows:

Step 1: Construct line l3 which will be perpendicular to line l1 from point A

Step 2: Construct line l2 which will be perpendicular to l3 through point A. (Hint: This is the same as bisecting a straight angle.)

A

l1 3. Here is another method for constructing a line parallel to a given line through a point not on the line, not using perpendicular lines. Using the construction for copying an angle, construct a line parallel to line L through point P.

P

L

4a) Construct the perpendicular bisector of BC. b) Construct the angle bisector of ≮B.

C

A

B

Sum it Up!!

A perpendicular bisector of a segment passes through the ______ofmidpoint the segment and forms ______right angles with the segment. (Mark the diagram to show this.) A A point A is said to be equidistant from two different points B and C if AB = AC. B C (Mark the diagram to show this.) Name ______CC Geometry H Date ______HW #9 1. Construct the perpendicular bisector of the segment below. A

B

2. Construct the line perpendicular to line l through point A. List all the steps necessary to complete the construction. A

l

3. Construct the perpendicular bisectors of AB, BC , and CA on the triangle below. What do you notice about the segments you have constructed?

A

B C 4. Two homes are built on a plot of land. Both homeowners have dogs, and are interested in putting up as much fencing as possible between their homes on the land, but in a way that keeps the fence equidistant from each home. Use your construction tools to determine where the fence should go on the plot of land.

5. How will the fencing alter with the addition of a third home?

Review 1. In ΔABC, ≮B = ≮C. If AB = x2 + 8x, BC = 3x + 5 and AC = 20, find the perimeter ΔABC.

2. In the diagram below, ≮BCA = 8x – 15, ≮DBC = 2x + 12 and ≮BDC = 4x + 9. Find m≮BCD.