AJ has been constructed as the angle bisector of BAD. B. AJ is the of BD.
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3 im #9: How do we construct a perpendicular bisector? Do Now: Using the angle below: 1. isect the angle. Label the bisector D. 2. Construct a copy of DC using vertex '. CC Geometry H C ' Relevant Vocabulary Two lines are PERPENDICULR ( ) 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. SEGMENT ISECTOR passes through the of a segment. n NGLE ISECTOR is a ray (line/segment) that divides an into 2 = or parts. J has been constructed as the angle bisector of D. D J Draw D. Label C, the point of intersection of D and J. Notice: (1) C = CD so C is a. Therefore J is a of D. (2) C = DC. Each of these angles measures Therefore, D and J are. J is the of D. 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.)
4 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.] linebisect.html Steps for Creating a Perpendicular isector. Construct the perpendicular bisector of CD. Label the midpoint M and label the perpendicular bisector as EF. Name one right angle:. C D
5 Relevant Vocabulary: EQUIDISTNT point is said to be equidistant from two different points and C if = C. Draw a diagram. CDE is the perpendicular bisector of. (C, D, and E are collinear points.) Using your compass, what conclusion can you make about the following pairs of segments? 1) C and C equal, circle C with rad. C goes through pt. 2) D and D equal, circle D with rad. D goes through pt. 3) E and E equal, circle E with rad. E goes through pt. ased on your findings, fill in the observation below. ny point on the perpendicular bisector of a line segment is from the endpoints of the line segment. Why? ny circle we constructed w/a center on the perp. bisector and a radius that passed through one endpt, also passed through the other endpt. ll 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. l X Step 1: Draw circle X so that the circle intersects line l in two points. Step 2: Label the two points of intersection as and C. Step 3: Draw circle : center, radius. Step 4: Draw circle C: center, radius. Step 5: Label the unlabeled intersection of circle and circle C as D. Step 6: Draw the perpendicular bisector: line.
6 Exercises 1. Divide segment into 4 segments of equal length. 2. Construct parallel lines l 1 and l 2 as follows: Step 1: Construct line l 3 which will be perpendicular to line l 1 from point Step 2: Construct line l 2 which will be perpendicular to l 3 through point. (Hint: This is the same as bisecting a straight angle.) l 1
7 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 C. b) Construct the angle bisector of. C Sum it Up!! perpendicular bisector of a segment passes through the of the segment and forms angles with the segment. (Mark the diagram to show this.) point is said to be equidistant from two different points and C if = C. (Mark the diagram to show this.) C
8 Name Date 1. Construct the perpendicular bisector of the segment below. CC Geometry H HW #9 2. Construct the line perpendicular to line l through point. List all the steps necessary to complete the construction. l 3. Construct the perpendicular bisectors of, C, and C on the triangle below. What do you notice about the segments you have constructed? C OVER
9 4. Two homes are built on a plot of land. oth 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 ΔC, = C. If = x 2 + 8x, C = 3x + 5 and C = 20, find the perimeter ΔC. 2. In the diagram below, C = 8x 15, DC = 2x + 12 and DC = 4x + 9. Find m CD.
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