explorations in euclidean geometry

eileen mooney, Sep 7, 2014

Table of Contents

  1. building relationships between points
    1. exploring the relationship between three points
    2. triangle inequality
    3. reflecting a segment across a line
    4. Circle as a set of equidistant points
    5. the equidistance of points in relation to a segment.
    6. equidistance in relation to two points
    7. exploration: equidistance and slope
    8. median-altitude relationship in different triangles
  2. constructing unique triangles
    1. side angle side
    2. side side side triangles
    3. two angles and a side in between
    4. ssa acute angle situation.
    5. ssa obtuse triangle situation
    6. ssa unique situation.
  3. parallel lines
    1. Same Side Interior
    2. Corresponding Angles
    3. angles around a transversal
    4. pile of points equidistant to a line
  4. similarity
    1. shapes of triangles
    2. triangle similarity, given side lengths
    3. similar parallelograms
    4. AAAA similarity?
    5. Bri's Conjecture
    6. angle bisector in a triangle

1.

building relationships between points

  • 1. exploring the relationship between three points
  • 2. triangle inequality
  • 3. reflecting a segment across a line
  • 4. Circle as a set of equidistant points
  • 5. the equidistance of points in relation to a segment.
  • 6. equidistance in relation to two points
  • 7. exploration: equidistance and slope
  • 8. median-altitude relationship in different triangles

1.1.

exploring the relationship between three points

Geogebra Exploration 1: Three points Problem Set 1. Play around with the application: drag A, B, and C around the plane. Click the boxes and see what they do. 2. Begin this exercise by first hiding the segments. Position the points in such a way where all three appear to be collinear. Then, click the boxes and record the results (given that the values are lengths of segments). 3. Unclick the boxes again. Now, move one point (easiest to move the middle one) so that the points are non-collinear. What relationship do the three points have now? 4. How far off the line did you have to move the point in order to establish this relationship? Discuss this with your group. Be as precise with your language as possible. 5. Freely click the boxes to help visualize the relationship. When you do, what do you notice about the lengths that are given? Record the results. 6. What is different about the results that you wrote from Question 2? 7. Express a general statement about the relationship between the lengths of the segments when the points are collinear. 8. Express a general statement about the relationship between the lengths of the segments in the second relationship.
– eileen mooney

1.2.

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1.5.

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1.6.

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1.7.

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1.8.

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2.

constructing unique triangles

  • 1. side angle side
  • 2. side side side triangles
  • 3. two angles and a side in between
  • 4. ssa acute angle situation.
  • 5. ssa obtuse triangle situation
  • 6. ssa unique situation.

2.1.

side angle side

If given two segments and an angle in between the given segments, can you determine a unique triangle? Why or why not? Discuss the relationship between the points. If you were given the same angle and the same length sides and asked to put them together in the same way, would you get the same triangle back? Why or why not? Let's say that we will call the orientation above as "SAS". What if the sides and the angle were oriented as SSA, so that the angle was not between the two sides? Would we be able to determine a triangle?
– eileen mooney

2.2.

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3.

parallel lines

  • 1. Same Side Interior
  • 2. Corresponding Angles
  • 3. angles around a transversal
  • 4. pile of points equidistant to a line

3.1.

Same Side Interior

Drag point D around and notice the angle measures changing. Drag D so that the lines appear to be as parallel as possible. Record the angle measures. Drag D so that the lines will eventually intersect on the right. Record the angle measures. Drag D so that the lines will eventually intersect on the left. Record the angle measures. Discuss the relationship between the angle measures and the "parallelness" of the lines. What seems to be the case?
– eileen mooney

3.2.

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3.4.

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4.

similarity

  • 1. shapes of triangles
  • 2. triangle similarity, given side lengths
  • 3. similar parallelograms
  • 4. AAAA similarity?
  • 5. Bri's Conjecture
  • 6. angle bisector in a triangle

4.1.

shapes of triangles

For two shapes to be similar, they have to be the same shape, but they do not have to be the same size. Drag the vertices of triangle DEF so that it is similar to triangle ABC, but not also the same size. Study the diagrams and make a few observations about the relationships between elements of the triangles. Share them with your group. Come to a consensus about your observations. What constitutes similarity? In order for a set of triangles to be similar, what relationships must be withheld amongst them?
– eileen mooney

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4.6.

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