Trapezoid/Isosceles Trapezoid

Construction of a trapezoid:

-To construct a trapezoid using GSP I started by clicking the icon for a line segment on the left side of the screen, and then I constructed a line segment. Next, I clicked the icon for a point and constructed a point above the line segment. I then constructed a line parallel to the line segment and through the point. Then, I simply connected the endpoints of the line segment to points on the line that was parallel to the line segment. The rough sketch of the trapezoid looked like this:

– From previous knowledge of trapezoids, I knew that they did not have some of the properties of the other shapes that I have explored, but I went through the same procedure as I did with the rectangle, rhombus, and square. I added in the measures of the side lengths, angles, diagonals, and the angles created by the diagonals.

– After examining the trapezoid and its measurements, I was able to find the following properties:

1. A trapezoid has two pair of angles that are supplementary.

2. The diagonals of a trapezoid divide themselves into proportional parts.

3. The diagonals of a trapezoid form angles that have congruent opposite angles.

4. A trapezoid has at least one pair of opposite parallel sides.

Construction of an Isosceles Trapezoid:

– To construct an isosceles triangle in GSP, I began by constructing a normal line segment. With the line segment highlighted, I went to the construct menu and selected “midpoint” to construct a midpoint on the line segment. I then constructed a perpendicular line to the line segment and going through the midpoint. After that, I connected a line segment to an endpoint of the original line segment. I highlighted the perpendicular line and went to the transform menu and selected “mark mirror” to set a place to translate a future figure over. After setting the mirror, I highlighted the newest line segment and went back to the transform menu and selected the command “reflect” to reflect the same line over the perpendicular line and connected to the opposite endpoint of the original line segment. I then constructed a line parallel to the original line segment and passing through the new endpoints. The rough sketch of the figure looked like this:

– In an isosceles trapezoid there are at least two congruent sides, so I predicted that it would have more properties that would make it similar to other figures than a regular trapezoid. I added in the measures of the side lengths, angles, diagonals, and angles created by the diagonals to determine the properties of the isosceles trapezoid below.

– From the sketch and analyzing all the measurements I was able to determine the following properties:

1. It has one pair of congruent opposite sides.

2. The diagonals are congruent.

3. The diagonals are divided into proportional parts.

4. Two pairs of consecutive angles are congruent.

5. It has two pair of consecutive angles that are supplementary.

6. The angles formed by the diagonals are congruent to their respective opposite angles.

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