Scalene Triangle and Points of Concurrency

Transcription

1 Scalene Triangle and Points of Concurrency Geometry Date: I. Create a Scalene Triangle 1 A. Group A 1. On a piece of patty paper, use a ruler to carefully draw a scalene obtuse triangle. Make the triangle as large as possible. 2. No congruent sides. 3. One angle greater than 90 o. B. Group B 1. On a piece of patty paper, use a ruler to carefully draw a scalene right triangle. Make the triangle as large as possible. 2. No congruent sides. 3. One angles equal to 90 o. C. Group C mathworld.wolfram.com/concurrent.html 1 mathworld.wolfram.com/scalenetriangle.html Geometry Scalene Triangle and Points of Concurrency Page 1

2 1. On a piece of patty paper, use a ruler to carefully draw a scalene acute triangle. Make the triangle as large as possible. 2. No congruent sides. 3. All angle less than 90 o. D. Label the vertices A, B,C. E. Keep this triangle as your master copy. Do not fold or write on this triangle. II. Circumcenter 2 B. Write Circumcenter at the top of the paper. C. Create the perpendicular bisector 3 of each side. 1. Fold AB such that A and B overlap. 2. Crease the paper through to the opposite side. 3. Repeat for the other two sides. D. Label the point of intersection of the three perpendicular bisectors as O. E. Create the circumcircle 1. Draw a circle with center O and radius OA. 2. The circumcenter is the center of the circle that inscribes the triangle. 3. Is the circle outside of the triangle, touching only at the vertices? 2 mathworld.wolfram.com/circumcenter.html 3 mathworld.wolfram.com/perpendicularbisector.html Geometry Scalene Triangle and Points of Concurrency Page 2

3 III. Centroid 4 B. Write Centroid at the top of the paper. C. Create the median 5 of each side. 1. Fold AB such that A and B overlap. 2. Place a small crease at the midpoint with your thumb. Do not move your thumb. 3. Roll the paper with your other thumb such that a crease is formed from the midpoint to C 6 4. Repeat for the other two sides. D. Label the point of intersection of the three medians as G. E. The centroid is the center of mass 7 of the triangle. 1. Trace the triangle and and the centroid onto a new sheet of patty paper. Put the original aside. 2. Use the ruler to trim the triangle out of the patty paper. 3. Balance the triangle on your pencil at the centroid. 4. Does the triangle balance? 4 mathworld.wolfram.com/trianglecentroid.html 5 mathworld.wolfram.com/trianglemedian.html 6 The median is an example of a cevian, a line segment which joins a vertex of a triangle with a point on the opposite side (or its extension), mathworld.wolfram.com/cevian.html 7 hyperphysics.phy-astr.gsu.edu/hbase/cm.html Geometry Scalene Triangle and Points of Concurrency Page 3

4 IV. Incenter 8 B. Write Incenter at the top of the paper. C. Create the angle bisector 9 of each vertex angle. 1. Fold ABC such that AB and BC overlap. 2. Use your thumb to put a crease at the vertex B. 3. Put the crease through to the opposite side. 4. Repeat for the other two vertex angles D. Label the point of intersection of the three angle bisectors as I. E. Create the incircle 1. Fold AB onto itself and roll, keeping the line on top of itself, until the crease goes through I. 2. Label the point on AB as M. 3. Draw a circle with center I and radius IM. 4. The incenter is the center of the inscribed circle 5. Is the circle inside of the triangle, tangent to the sides? 8 mathworld.wolfram.com/incenter.html 9 mathworld.wolfram.com/anglebisector.html Geometry Scalene Triangle and Points of Concurrency Page 4

5 V. Orthocenter 10 B. Write Orthocenter at the top of the paper. C. Create the altitude 11 of each side 1. Fold AB such that A and B overlap. Do not crease. 2. While keeping the line on top of itself, adjust the fold so that the crease goes to the vertex angle C. AB will overlap, but will not bisect. 3. Repeat for the other two sides. D. Label the point of intersection of the three attitudes as H. E. Orthocentroidal Circle Trace G from the paper labeled Centroid. 2. Create GH by folding a crease going through G and H. 3. Determine the midpoint of GH by folding it over on itself, such that G and H are on top of one another. 4. Mark the midpoint of GH as M. 5. Draw a circle with M as the center and MH as the radius 6. Circle M is the Orthocentroidal Circle. It is a central circle of the triangle, and contains the points G and H on it s circumference. 10 mathworld.wolfram.com/orthocenter.html 11 mathworld.wolfram.com/altitude.html 12 mathworld.wolfram.com/orthocentroidalcircle.html Geometry Scalene Triangle and Points of Concurrency Page 5

6 VI. de Longchamps Point 13 B. Write de Longchamps Point at the top of the paper. C. Trace, mark, and label the orthocenter H and the circumcenter O. D. Reflect the orthocenter. 1. Create HO by folding a crease going through H and O. 2. Fold the paper at point O so the line overlap. 3. Trace point H on the other side of the paper. E. Label as the de Longchamps Point L. F. Euler Line Trace, mark, and label the centroid G. 2. Trace, mark, and label the incenter I. 3. Draw a line containing the centroid, orthocenter, circumcenter and Longchamps point. 4. This is the Euler Line. 5. Does the incenter lie on this line? 13 mathworld.wolfram.com/delongchampspoint.html 14 mathworld.wolfram.com/eulerline.html Geometry Scalene Triangle and Points of Concurrency Page 6

7 VII. Consolidating points of concurrency B. Write Points of Concurrency, Isosceles, and what type of angle, on top of the paper. C. Trace, mark, and label the Circumcenter O. D. Trace, mark, and label the Centroid G. E. Trace, mark, and label the Incenter I. F. Trace, mark, and label the Orthocenter H. G. Trace, mark, and label the de Longchamps Point L. VIII. Summarize points of concurrency A. Assemble in a group consisting of one of each type of triangle. B. Trade triangles and trace triangles and points of concurrency. C. Discuss and develop a definition for each 1. Angle Bisector: 2. Incenter: 3. Perpendicular bisector: 4. Cicumcenter: 5. Median: 6. Centroid: 7. Altitude: 8. Orthocenter: Geometry Scalene Triangle and Points of Concurrency Page 7

8 D. Complete the table. Isosceles Acute Right Obtuse Centroid in/out On Euler Incenter in/out On Euler Circumcenter in/out On Euler Orthocenter in/out On Euler IX. Short answers. Complete on your own, in your notebook. A. How did the position of the points of concurrency change as we move from acute to obtuse vertex angles. For instance, were the points together, inside or outside of the triangle. Support your answer. B. What points of concurrency are on the Euler line, and what points were not? Did this change as the vertex angle changed? Support your answer. C. Points H and G always lie on the orthocentroidal circle. Does it seem that points H and G also always lie on the Euler line? If so, what is the segment of the Euler line inside the circle with respect to the orthecentroidal circle? Are there any points of concurrency that always lie inside the circle? Geometry Scalene Triangle and Points of Concurrency Page 8