You are probably thinking to yourself that I have made yet another typo, it should be CENTER not CENTERS, but you would be incorrect ( this time ). We understand that the center might mean the middle, but center is center, there is only one center. This is true for many things, but triangles have more than just one center ( they actually have quite a few ). We are going to look at the four major centers of a triangle Orthocenter, Centroid, Circumcenter, and Incenter. Each of these triangle centers is different and some are even useful!
Altitude of a Triangle: |
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Orthocenter of a Triangle: |
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Build the orthocenter at point D of ΔABC in Geogebra, for an Acute, Obtuse, & Right triangle |
Solution: Step: 1. Plot 3-ponts: A, B, C 2. Create segments AB, BC, AC 3. Create a perpendicular line from each segment through the point opposite it. 4. Create intersection points: - Point D at the intersection of the 3-perpendicular lines - Points at the inspections of the sides with the perpendicular lines 5. Create right the angle measurements Pont D will be the Orthocenter of ΔABC Press  or
to run the Geogebra build simulation.Watch the video for this example: |
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Quick Question... Instructions:
To use the app:
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Median of a Triangle:
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| The median of a triangle is
a segment whose endpoints are a vertex of the triangle and the
midpoint
of the opposite side. |
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Centroid of a Triangle:
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Build the centroid at point P of ΔABC in Geogebra, for an Acute, Obtuse, & Right triangle .
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Solution: Step: 1. Plot 3-ponts: A, B, C 2. Create segments AB, BC, AC 3. Create a midpoint on each segment, label each point as: - Midpoint X on segment BC - Midpoint Y on segment AC - Midpoint Z on segment AB 4. Create the 3-medians (segments) - Median AX - Median BY - Median CZ 5. Create point P at the intersections of the 3-medians Pont P will be the Centroid of ΔABC Press or
to run the Geogebra build simulation.Watch the video for this example: : |
Centroid Theorem:
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The
centroid of a triangle
is located the
distance from each vertex to the
midpoint of the
opposite side,
so in ΔABC: we
have the following:   |
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Use Geogebra to verify that the centroid of the triangle
is always
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We will continue with the Geogebra simulation from where we left off for
building the centroid
of the triangle. 1. Find the distance from A to X, A to P, & X to P AX = 3.37, AP = 2.24, & XP = 1.12  2. Find the distance from B to Y, B to P, & Y to P BY = 2.82, BP = 1.88, & YP = 0.94  3. Find the distance from C to Z, C to P, & Z to P CZ = 4.08, CP = 2.72, & ZP = 1.36  Calculator Screen Output:  Press or
to run the Geogebra build simulation.Watch the video for this example: |
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Quick Question... Instructions:
To use the app:
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Circumcenter of a Triangle:
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Build the circumcenter at point
G of ΔABC in Geogebra, for an Acute, Obtuse, & Right triangle
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Solution: Step: 1. Plot 3-ponts: A, B, C 2. Create segments AB, BC, AC 3. Create a perpendicular bisector on each side of the triangle 4. Create point G at the intersections of the 3-perpendicular bisectors. 5. Create right the angle measurements 6. Create segments from each vertex to the circumcenter. - segment AG - segment BG - segment CG 7. Create a circle with center G, through either point A, B, or C. Pont G will be the circumcenter of ΔABC Press or
to run the Geogebra build simulation.Watch the video for this example: |
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Quick Question... Instructions:
To use the app:
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Incenter of a triangle:
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Build the incenter at point
N of ΔABC in Geogebra, for an Acute, Obtuse, & Right triangle
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Solution: Step: 1. Plot 3-ponts: A, B, C 2. Create segments AB, BC, AC 3. Create the angle bisector for each vertex 4. Create point N at the intersections of the 3-angle bisectors. 5. Create a perpendicular line from each side trough point N 6. Create an intersection point on each side, with the line perpendicular to it, label each point as: - X on side BC - Y on side AC - Z on side AB 7. Create the 3-segments - NX - NY - NZ 8. Create right the angle measurements 9. Create a circle with center N, through either point X, Y, or Z.. Pont G will be the circumcenter of ΔABC Press or
to run the Geogebra build simulation.Watch the video for this example: |
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Quick Question... Instructions:
To use the app:
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A. Create ΔHJK, where  J
= 90o, and identify its
orthocenter. |
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B. Create equilateral ΔABC, where M is the midpoint of
 ,N is the midpoint of  ,
and L is the midpoint of
 Then identify the centroid of the circle as point P. Lastly, adjust the size of ΔABC so that  =
2What is the measure of  ? |
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| C. Create ΔRST so that the circumcenter is
outside of the triangle. |
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| D. Create ΔUVW and identify the incenter. |
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| 1. Create ΔABC, where
= 3,
= 4,
= 5. Identify its Orthocenter as point P. |
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| 2. Create ΔABC, where
= 6,
= 6,
= 6. Identify its Orthocenter as point P. |
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| 3. Create ΔABC, where
= 7,
= 8,
= 14. Identify its Orthocenter as point P. |
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| 4. In ΔABC Z is the midpoint of
,
Y is the midpoint of
, and X is the midpoint of with
its Centroid at point P.If 
= 2, what is the length of
= ? |
Figure for problems 4, 5, & 6 |
| 5. In ΔABC Z is the midpoint of
,
Y is the midpoint of
, and X is the midpoint of with
its Centroid at point P.If 
= 1, what is the length of
= ? |
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| 6. In ΔABC Z is the midpoint of
,
Y is the midpoint of
, and X is the midpoint of with
its Centroid at point P.If 
= 9, what is the length of
= ? |
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| 7. Create ΔABC, where
= 6,
= 8,
= 10. Identify its Circumcenter as point P. |
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| 8. Create ΔABC, where
= 6,
= 5,
= 4. Identify its Circumcenter as point P. |
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| 9. Create ΔABC, where
= 16,
= 8,
= 10. Identify its Circumcenter as point P. |
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| 10. Create ΔABC, where
= 9,
= 12,
= 15. Identify its Incenter as point P. |
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| 11. Create ΔABC, where
= 9,
= 11,
= 10. Identify its Incenter as point P. |
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12. Create ΔABC, where
= 9,
= 22,
= 15. Identify its Incenter as point P. |
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13. A lamp post is to be placed in a park so that it will provide
light for 3-park benches. The park benches form a triangle, where bench #1 is 170 yards from bench #2 and 150 yards from bench #3. Benches #2 & #3 are 200 yards apart. If the lamp post is placed so that it is the same distance from each bench and the light used on the post will illuminate a area with a radius of 125 yards. Will this light provide enough light for the 3-park benches? Note: Draw this situation out in the Jamboard and use Geogebra to help you answer this question. |
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| 14. The following floor truss design is
being used in the construction of a new house. The HVAC contractor needs to run flexible ducting through the truss. The contractor has the option of buying 3" diameter, 4"diameter, or 6" diameter ducting. If the truss is made up of equilateral triangles with a base of 8 inches, what will be the largest outside diameter ducting that the contractor could use? Note: Draw this situation out in the Jamboard and use Geogebra to help you answer this question. |
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15. Create an acute scalene triangle and identify the
Orthocenter,
Centroid, and Circumcenter ( make sure you hide the lines used to create each triangle center after you made it and use a different color for each of the center.). a. Take a screenshot of your triangle and paste it into the Jamboard b. Do you notice anything about these 3-points? c. Next create a line using any two of the points and move the triangle around to see what happens. Take a screenshot of this triangle and paste it into the Jamboard d. Did you notice anything about these 3-points? e. If you created the Incenter for this triangle where do you expect the point to land? f. Create the Incenter and see where it lands. Take a screenshot of this triangle and paste it into the Jamboard g. Did it land where you expected it to land? h. Why do you think it landed where it did? i. Do you think there might be other points on this line based on a triangle? |
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| 16. Check this out: Exeter Point |
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