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1 Lesson 3: Copy and Bisect and Angle Lesson 4: Construct a Perpendicular Bisector Lesson 5: Points of Concurrencies Student Outcomes: ~Students learn how to bisect an angle as well as how to copy an angle ~Students construct a perpendicular bisector and discover the relationship between symmetry with respect to a line and a perpendicular bisector ~Students become familiar with vocabulary regarding two points of concurrencies and understand why the points are concurrent. Definitions: Definitions: Angle Equidistant Midpoint Right Angle Zero and Straight Angle Perpendicular Homework: #2 due next class Definitions: Concurrent Point of concurrency Circumcenter of a triangle Perpendicular Bisector Equidistant Incenter Jun 16 11:17 AM Which angle is bigger? Exterior Region Interior Region Aug 8 8:46 PM 1
2 Consider the following ideas: ~are angles the only geometric figures that can be bisected? ~segments can also be bisected ~a line of reflection must exist so that when the figure is folded along this line, each point on one side maps with another point on the other side ~a ray can't be bisected ~what determines wether a figure can be bisected? ~what kinds of figures cannot be bisected? Aug 8 8:52 PM To bisect an angle: 1. Label vertex of angle as A 2. Draw circle A of any size of radius 3. Label intersections of circle A with rays of angle as B and C 4. Adjust your compass to the length between points B and C and draw circles B and C 5. Label the intersections of circles B and C in the interior of the angle as D 6. Draw ray AD A B C D Aug 8 9:28 PM 2
3 Bisect the angle: Aug 8 10:06 PM Steps to copy an angle: 1. Label the vertex of the original angle B 2. Draw new ray EG as one side of the angle to be drawn 3. Draw circle B of any size radius 4. Label the intersections of the radius and the angle as A and C 5. Draw circle E, with the same size radius as circle B 6. Label intersection of circle E with ray EG as F 7. Adjust compass to the distance from A to C and make a circle at F 8. Label either intersection of circles E and F as D 9. Draw ray ED B E A F C G Aug 8 9:35 PM 3
4 Copy the angle: Aug 20 4:42 PM Aug 20 4:43 PM 4
5 To construct a perpendicular bisector: 1. Label the endpoints of the segment as A and B 2. Adjust compass so that the length is just slightly more than half of the line segment 3. Draw two congruent circles at point A and B 4. Mark intersections as C and D 5. Draw line CD C A B D Aug 8 9:58 PM Points of Concurrency of a Triangle Medians = Centroid Altitudes = Orthocenter Angle Bisectors = Incenter Perpendicular Bisectors = Circumcenter Mar 8 10:22 AM 5
6 Incenter The point where three angle bisectors meet Mar 8 10:30 AM Circumcenter The point where the three perpendicular bisectors of the sides of a triangle meet Mar 8 10:30 AM 6
7 The centroid of a triangle is created by the medians (a line segment whose endpoints are a vertex of a triangle and the midpoint of the opposite side). To make a centroid, bisect each side and connect the midpoint with the opposite vertex. B A Z M Y X C AM = 2/3AX OR AM = 2*MX Every triangle has three medians The point of concurrency of the medians is the centroid of the triangle. The centroid of a triangle is located 2/3 of the distance from each vertex to the midpoint of the opposite side. Aug 20 4:49 PM Sep 12 12:56 PM 7
8 Sep 12 12:56 PM The median is split into a ratio of 2:1. E is the centroid of triangle ABC. If AD is 12, find the value of AE and ED. C E D A B The larger part of the median is twice the smaller. E is the centroid of triangle ABC. If AE is 4, find the value of ED. E is the centroid of triangle ABC. If ED is 8, find the value of AE. Mar 21 1:17 PM 8
9 An orthocenter is made with the altitudes of the triangle (a line segment that connects a vertex to its opposite side and is perpendicular to that side). Every triangle has three altitudes. The altitude may be on the inside or the outside of the triangle. The point of concurrency of the altitudes is called the orthocenter. Aug 20 4:54 PM Locus a set of points that satisfies a given condition There are five types of simple locus. Jan 21 1:06 PM 9
10 1. The locus of points a distance from a point results in a circle. Example: Find the locus of points 2 units away from (1, 3). Jan 21 1:07 PM 2. The locus of points a distance from a line results in two parallel lines. Example: Find the locus of points 3 units away from y = 2x + 1. Jan 21 1:08 PM 10
11 3. The locus of points equidistant from two points results in a line (the perp bisector of the line segment joining the points). Example: Find the locus of points equidistant from (1, 3) and (1, 5). Jan 21 1:10 PM 4. The locus of points equidistant from two parallel lines results in one line. Example: Find the locus of points equidistant from y = 3 and y = 7. Jan 21 1:10 PM 11
12 5. The locus of points equidistant from two intersecting lines results in two lines that bisect the angles formed by the lines. Example: Find the locus of points equidistant from y = x + 1 and y = x 1. Jan 21 1:12 PM 12
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Name: Triangles Test Type of Triangle Definition Drawing Right Obtuse Acute Scalene Isosceles Equilateral Number of congruent angles = Congruent sides are of the congruent angles Name the triangles below,