7.2: Properties of Trapezoids and Isosceles Trapezoids
This activity is intended to supplement Geometry, Chapter 6, Lesson 5.
In this activity, you will explore:
- the properties of trapezoids, including isosceles trapezoids
To begin, open up a blank Cabri Jr. document.
Let’s start with a line segment, \begin{align*}AB\end{align*}, and a point \begin{align*}C\end{align*} above the segment.
Create a line through \begin{align*}C\end{align*} that is parallel to \begin{align*}\overline{AB}\end{align*}. Construct a point on the new line and label it \begin{align*}D\end{align*}.
Hide the parallel line and complete the trapezoid. Measure the interior angles. Are the base angles at \begin{align*}C\end{align*} and \begin{align*}D\end{align*} congruent? Will they ever be? What about the base angles at \begin{align*}A\end{align*} and \begin{align*}D\end{align*}? Measure the lengths of the diagonals \begin{align*}\overline{AD}\end{align*} and \begin{align*}\overline{BC}\end{align*}. Will they ever be congruent?
To verify that the lines are parallel, construct lines through \begin{align*}C\end{align*} and \begin{align*}D\end{align*} that are perpendicular to \begin{align*}\overline{AB}\end{align*}.
Construct the points of intersection of the new lines with \begin{align*}\overline{AB}\end{align*}. Construct lines segments to connect \begin{align*}C\end{align*} and \begin{align*}D\end{align*} to the line through \begin{align*}\overline{AB}\end{align*} and measure the lengths of these segments. Will these segments always be equal? Should they be? Why?
Label the new points on \begin{align*}\overline{AB}\end{align*} as \begin{align*}E\end{align*} and \begin{align*}F\end{align*} and measure the lenghts of \begin{align*}\overline{AC}\end{align*} and \begin{align*}\overline{BD}\end{align*}. Will \begin{align*}\overline{AC}\end{align*} ever be congruent to \begin{align*}\overline{BD}\end{align*}?
Try to drag point \begin{align*}C\end{align*} or point \begin{align*}D\end{align*} to make \begin{align*}AC = BD\end{align*}. Due to the screen resolution, this can be very difficult. Construct \begin{align*}\overline{AE}\end{align*} and \begin{align*}\overline{BF}\end{align*}. For an isosceles trapezoid, these segments should also be congruent. Can you explain why?
In order to construct an isosceles trapezoid, start with a line segment, \begin{align*}AB\end{align*}. Construct the midpoint, \begin{align*}M\end{align*}, and another point, \begin{align*}P\end{align*}, on \begin{align*}\overline{AB}\end{align*}. Construct a line through the \begin{align*}M\end{align*} that is perpendicular to \begin{align*}\overline{AB}\end{align*}.
Press TRACE/F4 and select the Reflection option. Click on the perpendicular line through \begin{align*}M\end{align*} and then point \begin{align*}P\end{align*}.
A new point will appear on the line segment on the right side. Label this point \begin{align*}Q\end{align*}.
Construct perpendicular lines through \begin{align*}P\end{align*} and \begin{align*}Q\end{align*}. Construct point \begin{align*}C\end{align*} on the perpendicular through \begin{align*}P\end{align*}.
Construct a perpendicular to \begin{align*}\overline{AB}\end{align*} through point \begin{align*}Q\end{align*} and a line through \begin{align*}C\end{align*} that is parallel to \begin{align*}\overline{AB}\end{align*}. Construct point \begin{align*}D\end{align*} at the intersection of these two lines.
Hide the parallel and perpendicular lines and points \begin{align*}M\end{align*}, \begin{align*}P\end{align*} and \begin{align*}Q\end{align*}. Complete the trapezoid and measure \begin{align*}\overline{AC}\end{align*} and \begin{align*}\overline{BD}\end{align*}. Can you explain why these line segments are congruent? In an isosceles trapezoid, the diagonals are congruent and the adjacent base angles are congruent.
Measure the base angles—the interior angles at \begin{align*}A\end{align*}, \begin{align*}B\end{align*}, \begin{align*}C\end{align*} and \begin{align*}D\end{align*}. Which angles are congruent? Did you expect those angles to be congruent? Can you prove that they are?
Construct and measure the lengths of the two diagonals \begin{align*}\overline{AD}\end{align*} and \begin{align*}\overline{BC}\end{align*}. Should they be congruent? Can you prove that they are?
Drag point \begin{align*}C\end{align*} and watch the angles and sides. Are all of the properties established above preserved?
However, one property that is common to all trapezoids is that the line segment connecting the non-parallel sides is also parallel to these sides and its length is half the sum of the parallel sides. Construct a trapezoid \begin{align*}ABCD\end{align*}. Construct the midpoints at \begin{align*}E\end{align*} and \begin{align*}F\end{align*} and construct \begin{align*}\overline{EF}\end{align*}.
Measure the lengths of \begin{align*}\overline{CD}\end{align*}, \begin{align*}\overline{EF}\end{align*} and \begin{align*}\overline{AB}\end{align*}. How could you prove that \begin{align*}\overline{EF}\end{align*} is parallel to \begin{align*}\overline{CD}\end{align*} and \begin{align*}\overline{AB}\end{align*}?
Use the Calculate tool to find the sum of the lengths of \begin{align*}\overline{CD}\end{align*} and \begin{align*}\overline{AB}\end{align*}.
Place the number “2” on the screen and divide the sum by 2 using the calculate tool. Will this result always equal the length of \begin{align*}\overline{EF}\end{align*}? Can you prove this result?