Points of Concurrency in Triangles
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1 Grade level: 9-12 Points of Concurrency in Triangles by Marco A. Gonzalez Activity overview In this activity, students will use their Nspire handhelds to discover the different points of concurrencies in triangles. The students will take advantage of the dynamic capabilities of this very unique handheld to discover the circumcenter, incenter, and centroid of triangles. Concepts Triangles, points of concurrency, circumcenter, incenter, centroid, perpendicular bisectors, angle bisectors, medians Teacher preparation This activity is designed for use in a Geometry classroom. Prior to beginning this activity, students should have prior knowledge of perpendicular bisectors, angle bisectors, medians, vertices of triangles, and the different types of triangles. Classroom management tips This activity is intended to be a teacher-led activity in which students will make their own discoveries pertaining to the points of concurrency of triangles. Students should be encouraged to make predictions of how the points of concurrency will behave if the vertices of the triangle are translated. TI-Nspire Applications Graphs and Geometry Page Step-by-step directions The following step-by-step directions are the same as the student directions. Circumcenter First, open a Graphs and Geometry page by Figure 1 Your screen should be a blank screen. Points of Concurrency in Triangles by Marco A. Gonzalez Page 1
2 Grade level: 9 th 12th Using the NavPad, move the cursor to a lower left part of the screen and press a. Before moving the cursor, label the point by pressing the letter A. pressing C. Press d. (see figure 1) Now let s calculate the perpendicular bisectors of each of the sides of the triangle. Press b, 9: Construction, 3: Perpendicular Bisector. Using the NavPad, move the cursor to side AB and press. Using the NavPad, move the cursor to side BC and press. Now, move the cursor to side AC and press. Press d. (see figure 2) Figure 2 Figure 3 find the point of intersection between the perpendicular bisectors. To do this, press b, 6 : Points & Lines, 3 : Intersection Points. Move the NavPad to any of the perpendicular bisectors and press a. Move the cursor to another perpendicular bisector and press a. Before you move the cursor press Q. Now let s hide the perpendicular bisectors. Move the cursor to one of the perpendicular bisectors and press /b. Scroll down or simply press 3: Hide/Show. Repeat the process for the other two perpendicular bisectors. (see figure 3) Figure 4 Point Q is called the circumcenter of the triangle. Now press b, 7: Measurement, 1: Length. Move the cursor to point A and press a. Now move the cursor to point Q and press a. There should be an ghost image of a number. Move the image to a free part of the screen and lock the number in place by pressing a. Move the cursor to point B and press a. Now move the cursor to point Q and press a. There should be a ghost image of a number. Move the image below the first measurement number and lock it in place by pressing a. Move the cursor to point C and press a. Now move the cursor to point Q and press a. There should be a ghost image of a number. Move the image below the other two measurements and lock it in place by pressing the a. (see figure 4) Figure 5 Page 2
3 Grade level: 9 th 12th until the point is grabbed. Now translate the point and notice the measurements on the screen. (see figure 5) What conjecture can you make about the circumcenter of a triangle? Incenter Open a new Graphs and Geometry page by Figure 6 Your screen should be a blank screen. 6) (see figure Figure 7 Using the NavPad, move the cursor to any part of the screen and press a. Before moving the cursor, label the point by pressing the letter A. pressing C. Press d. (see figure 7) Now press 9: Construction, 4: Angle Bisector. Move the cursor to point B and press a, move the cursor to point A and press a, move the cursor to point C and press a. A segment bisecting angle A should be drawn. (see figure 8) Repeat the process for the other two angles. After all three angle bisectors are drawn, press d. (see figure 9) Figure 8 Page 3
4 Grade level: 9 th 12th find the point of intersection between the angle bisectors. To do this, press b, 6 : Points & Lines, 3 : Intersection Points. Move the NavPad to any of the angle bisectors and press a. Move the cursor to another angle bisector and press a. Before you move the cursor press Q. Now let s hide the angle bisectors. Move the cursor to one of the angle bisectors and press /b. Scroll down or simply press 3: Hide/Show. Repeat the process for the other two angle bisectors. (see figure 10) Point Q is called the Incenter of the triangle. Figure 9 until the point is grabbed. Now translate the point. What do you notice? Extension Draw a perpendicular segment from the Incenter to each of the sides of the triangle. Find a point of intersection between the line segment and each of the sides. Now measure the distance between the Incenter and each of the points of intersection on the sides of the triangle. What do you notice? Figure 10 Centroid Open a new Graphs and Geometry page by Figure 11 Page 4
5 Grade level: 9 th 12th Your screen should be a blank screen. Using the NavPad, move the cursor to any part of the screen and press a. Before moving the cursor, label the point by pressing the letter A. pressing C. Press d. (see figure 11) Now, let s determine the midpoints of each of the sides of the triangle. Press b, 9: Construction, 5: Midpoint. Move the cursor to point A of the triangle and press a. Move the cursor to point B and press a. Since the cursor is already at point B, press a and move the cursor to point C. Press a. Since the cursor is already at point C, press a and move the cursor back to point A. Press a. Press d. You triangle should have a midpoint in each of its sides. (see figure 12) Figure 12 Figure 13 Now, we need to draw the medians of the triangle. Press b, 6: Points & Lines, 5: Segment. Move the cursor to point A on the triangle and press a. Now move the cursor to the midpoint of the side opposite vertex A of the triangle and press a. Move the cursor to point B on the triangle and press a. Now move the cursor to the midpoint of the side opposite vertex B of the triangle and press a. Move the cursor to point C on the triangle and press a. Now move the cursor to the midpoint of the side opposite vertex C of the triangle and press a. (see figure 13) find the point of intersection between the medians of the triangle. To do this, press b, 6 : Points & Lines, 3 : Intersection Points. Move the NavPad to any of the three medians and press a. Move the cursor to another median and press a. Before you move the cursor press Q. Figure 14 Page 5
6 Grade level: 9 th 12th Now let s hide the medians. Move the cursor to one of the medians and press /b. Scroll down or simply press 3: Hide/Show. Repeat the process for the other two medians. (see figure 14) Point Q is called the Centroid of the triangle. Note: The centroid of a triangle is known as the center of gravity because this is where the triangle will balance. until the point is grabbed. Now translate the point. What do you notice? Activity extensions The student should be encouraged to make predictions during the process of this activity. Students should be encouraged to explore other possible extensions of points of concurrencies (e.g. orthocenter, etc.) If time permits, the student should be encouraged to experiment and attempt to apply these points of concurrency to other polygons. Student TI-Nspire Document Circumcenter First, open a Graphs and Geometry page by Figure 1 Page 6
7 Grade level: 9th 12th Your screen should be a blank screen. Using the NavPad, move the cursor to a lower left part of the screen and press a. Before moving the cursor, label the point by pressing the letter A. pressing C. Press d. (see figure 1) Figure 2 Now let s calculate the perpendicular bisectors of each of the sides of the triangle. Press b, 9: Construction, 3: Perpendicular Bisector. Using the NavPad, move the cursor to side AB and press. Using the NavPad, move the cursor to side BC and press. Now, move the cursor to side AC and press. Press d. (see figure 2) find the point of intersection between the perpendicular bisectors. To do this, press b, 6 : Points & Lines, 3 : Intersection Points. Move the NavPad to any of the perpendicular bisectors and press a. Move the cursor to another perpendicular bisector and press a. Before you move the cursor press Q. Figure 3 Now let s hide the perpendicular bisectors. Move the cursor to one of the perpendicular bisectors and press /b. Scroll down or simply press 3: Hide/Show. Repeat the process for the other two perpendicular bisectors. (see figure 3) Point Q is called the circumcenter of the triangle. Now press b, 7: Measurement, 1: Length. Move the cursor to point A and press a. Now move the cursor to point Q and press a. There should be an ghost image of a number. Move the image to a free part of the screen and lock the number in place by pressing a. Move the cursor to point B and press a. Now move the cursor to point Q and press a. There should be a ghost image of a number. Move the image below the first measurement number and lock it in place by Figure 4 Page 7
8 Grade level: 9th 12th pressing a. Move the cursor to point C and press a. Now move the cursor to point Q and press a. There should be a ghost image of a number. Move the image below the other two measurements and lock it in place by pressing the a. (see figure 4) until the point is grabbed. Now translate the point and notice the measurements on the screen. (see figure 5) What conjecture can you make about the circumcenter of a triangle? Figure 5 Incenter Open a new Graphs and Geometry page by Figure 6 Your screen should be a blank screen.(see figure 6) Using the NavPad, move the cursor to any part of the screen and press a. Before moving the cursor, label the point by pressing the letter A. pressing C. Press d. (see figure 7) Now press 9: Construction, 4: Angle Bisector. Move the cursor to point B and press a, move the cursor to point A and press a, move the cursor to point C and press a. A segment bisecting angle A should be drawn. (see figure 8) Repeat the process for the other two angles. After all three Figure 7 Page 8
9 Grade level: 9th 12th angle bisectors are drawn, press d. (see figure 9) find the point of intersection between the angle bisectors. To do this, press b, 6 : Points & Lines, 3 : Intersection Points. Move the NavPad to any of the angle bisectors and press a. Move the cursor to another angle bisector and press a. Before you move the cursor press Q. Now let s hide the angle bisectors. Move the cursor to one of the angle bisectors and press /b. Scroll down or simply press 3: Hide/Show. Repeat the process for the other two angle bisectors. (see figure 10) Point Q is called the Incenter of the triangle. Figure 8 until the point is grabbed. Now translate the point. What do you notice? Extension Draw a perpendicular segment from the Incenter to each of the sides of the triangle. Find a point of intersection between the line segment and each of the sides. Now measure the distance between the Incenter and each of the points of intersection on the sides of the triangle. What do you notice? Figure 9 Centroid Figure 10 Open a new Graphs and Geometry page by Page 9
10 Grade level: 9th 12th Your screen should be a blank screen. Using the NavPad, move the cursor to any part of the screen and press a. Before moving the cursor, label the point by pressing the letter A. pressing C. Press d. (see figure 11) Figure 11 Now, let s determine the midpoints of each of the sides of the triangle. Press b, 9: Construction, 5: Midpoint. Move the cursor to point A of the triangle and press a. Move the cursor to point B and press a. Since the cursor is already at point B, press a and move the cursor to point C. Press a. Since the cursor is already at point C, press a and move the cursor back to point A. Press a. Press d. You triangle should have a midpoint in each of its sides. (see figure 12) Now, we need to draw the medians of the triangle. Press b, 6: Points & Lines, 5: Segment. Move the cursor to point A on the triangle and press a. Now move the cursor to the midpoint of the side opposite vertex A of the triangle and press a. Move the cursor to point B on the triangle and press a. Now move the cursor to the midpoint of the side opposite vertex B of the triangle and press a. Move the cursor to point C on the triangle and press a. Now move the cursor to the midpoint of the side opposite vertex C of the triangle and press a. (see figure 13) find the point of intersection between the medians of the triangle. To do this, press b, 6 : Points & Lines, 3 : Intersection Points. Move the NavPad to any of the three medians and press a. Move the cursor to another median and press a. Before you move the cursor press Q. Figure 12 Figure 13 Page 10
11 Grade level: 9th 12th Now let s hide the medians. Move the cursor to one of the medians and press /b. Scroll down or simply press 3: Hide/Show. Repeat the process for the other two medians. (see figure 14) Point Q is called the Centroid of the triangle. Note: The centroid of a triangle is known as the center of gravity because this is where the triangle will balance. until the point is grabbed. Now translate the point. What do you notice? Figure 14 Page 11
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