Special Segments of a Triangle

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1. Define Altitude:

Construct Acute Triangle ABC.

Choose vertex B and segment AC, construct "perpendicular line".

Choose this line and segment AC, construct "point at intersection", label as D.

Measure = and = .

Make the triangle right, with being the right angle, and then make the triangle obtuse, with being the obtuse angle.

Describe the location of the altitude in the acute, right, and obtuse triangles.

 Acute Triangle Right Triangle Obtuse Triangle

With the acute triangle, construct the altitudes from and .

What can you conclude about the three altitudes?

Choose two of the three altitudes and construct "point at intersection". Label this point "orthocenter".

Make the triangle right, with being the right angle, then make the triangle obtuse, with being the obtuse angle.

Describe the location of the orthocenter in the acute, right, and obtuse triangles.

 Acute Triangle Right Triangle Obtuse Triangle

2. Define Median:

Construct Acute Triangle ABC.

Choose segment BC, construct "point at midpoint", label as M.

Choose point M and point A, construct "segment".

Choose points B and M, construct "segment", choose points M and C, construct "segment".

Measure BM = and CM =

Make the triangle right, with being the right angle, and then make the triangle obtuse, with being the obtuse angle.

Describe the location of the median in the acute, right, and obtuse triangles.

 Acute Triangle Right Triangle Obtuse Triangle

With the acute triangle, construct the altitudes from and .

What can you conclude about the three medians?

Choose two of the three medians and construct "point at intersection". Label this point "centroid".

This point is considered to be the "center of gravity" of the triangle. Explain in you own words what that means.

Make the triangle right, with being the right angle, and then make the triangle obtuse, with being the obtuse angle.

Describe the location of the centriod in the acute, right, and obtuse triangles.

 Acute Triangle Right Triangle Obtuse Triangle

3. Define Perpendicular Bisectors:

Construct Acute Triangle ABC.

Choose segment AB, construct "point at midpoint", label as D.

Choose point D and segment AB, construct "perpendicular line".

Choose this perpendicular line, construct "point on object", label as E.

Measure = , = , AD = , DB =

Make the triangle right, with being the right angle, and then make the triangle obtuse, with being the obtuse angle.

Describe the location of the perpendicular bisector in the acute, right, and obtuse triangles.

 Acute Triangle Right Triangle Obtuse Triangle

Choose points A and E, measure distance:

Choose points B and E, measure distance:

Move point E along the perpendicular bisector to three different places and write down the distances from A to E and B to E.

AE = AE = AE =

BE = BE = BE =

A point that lies on the perpendicular bisector of a segment is from the of the segment.

With the acute triangle, construct the perpendicular bisectors from BC and AC.

What can you conclude about the three perpendicular bisectors?

Choose two of the three perpendicular bisectors and construct, "point at intersection". Label this point "circumcenter".

Make the triangle right, with being the right angle, and then make the triangle obtuse, with being the obtuse angle.

Describe the location of the circumcenter in the acute, right, and obtuse triangles.

 Acute Triangle Right Triangle Obtuse Triangle

Choose the "circumcenter" and Vertex B, construct "segment".

Choose the "circumcenter" and this new segment, construct "circle by center-radius".

Describe the result:

Make the triangle obtuse and right, is the result the same?

4. Define Angle Bisector:

Construct Acute Triangle ABC.

Choose , construct "angle bisector".

Choose this angle bisector and construct "point on object", label as E.

Measure = = =

Make the triangle right, with being the right angle, and then make the triangle obtuse, with being the obtuse angle.

Describe the location of the angle bisector in the acute, right, and obtuse triangles.

 Acute Triangle Right Triangle Obtuse Triangle

With the acute triangle, construct the angle bisectors from and .

What can you conclude about the three angle bisectors?

Choose two of the three angle bisectors and construct "point at intersection". Label this point "incenter".

Make the triangle right, with being the right angle, and then make the triangle obtuse, with being the obtuse angle.

Describe the location of the incenter in the acute, right, and obtuse triangles.

 Acute Triangle Right Triangle Obtuse Triangle

Choose the "incenter" segment BC, construct "perpendicular line".

Choose this perpendicular line and segment BC, construct "point of intersection".

Choose this point of intersection and the "incenter", construct segment.

Choose this segment and the "incenter", construct "circle by center-radius".

Describe the result:

Make the triangle obtuse and right. Does the result remain the same?

Construct Acute Triangle DEF.

Construct the angle bisector of .

Choose the angle bisector and construct "point on object", label as G.

Choose point G and segment DE, construct "perpendicular line".

Choose this line and segment DE, construct "point at intersection", label as X.

Construct segment XG, hide line XG.

Choose point G and segment FE, construct "perpendicular line".

Choose this line and segment FE, construct "point at intersection", label as Y.

Construct segment YG, hide line YG.

Measure XG = and YG = .

Move point G along the angle bisector to three different places and write down the lengths of segment XG and YG.

XG = XG = XG =

YG = YG = YG =

A point that lies along the bisector of an angle is from the of the angle.