Answer:
the quadrilateral ABCD is a trapezium (since one pair of opposite sides, BC and DA, are parallel) but not a parallelogram (since opposite sides AB and CD are not parallel and opposite sides BC and DA are not equal in length).
Step-by-step explanation:
To prove that the quadrilateral ABCD is a trapezium but not a parallelogram, we need to show that at least one pair of opposite sides are parallel, and at least one pair of opposite sides are not equal in length.
Let's calculate the position vectors for the sides AB, BC, CD, and DA using the given position vectors for the points A, B, C, and D.
1. Position vector for AB:
[tex]\[ \overrightarrow{AB} = \overrightarrow{B} - \overrightarrow{A} = \begin{bmatrix} -210 \\ 530 \end{bmatrix} - \begin{bmatrix} 410 \\ 710 \end{bmatrix} = \begin{bmatrix} -210 - 410 \\ 530 - 710 \end{bmatrix} = \begin{bmatrix} -620 \\ -180 \end{bmatrix} \][/tex]
2. Position vector for BC:
[tex]\[ \overrightarrow{BC} = \overrightarrow{C} - \overrightarrow{B} = \begin{bmatrix} -340 \\ -310 \end{bmatrix} - \begin{bmatrix} -210 \\ 530 \end{bmatrix} = \begin{bmatrix} -340 + 210 \\ -310 - 530 \end{bmatrix} = \begin{bmatrix} -130 \\ -840 \end{bmatrix} \][/tex]
3. Position vector for CD:
[tex]\[ \overrightarrow{CD} = \overrightarrow{D} - \overrightarrow{C} = \begin{bmatrix} 590 \\ -40 \end{bmatrix} - \begin{bmatrix} -340 \\ -310 \end{bmatrix} = \begin{bmatrix} 590 + 340 \\ -40 + 310 \end{bmatrix} = \begin{bmatrix} 930 \\ 270 \end{bmatrix} \][/tex]
4. Position vector for DA:
[tex]\[ \overrightarrow{DA} = \overrightarrow{A} - \overrightarrow{D} = \begin{bmatrix} 410 \\ 710 \end{bmatrix} - \begin{bmatrix} 590 \\ -40 \end{bmatrix} = \begin{bmatrix} 410 - 590 \\ 710 - (-40) \end{bmatrix} = \begin{bmatrix} -180 \\ 750 \end{bmatrix} \][/tex]
Now, let's analyze these vectors:
[tex]- AB: \( \begin{bmatrix} -620 \\ -180 \end{bmatrix} \)[/tex]
[tex]- BC: \( \begin{bmatrix} -130 \\ -840 \end{bmatrix} \)[/tex]
[tex]- CD: \( \begin{bmatrix} 930 \\ 270 \end{bmatrix} \)[/tex]
[tex]- DA: \( \begin{bmatrix} -180 \\ 750 \end{bmatrix} \)[/tex]
We see that AB and CD are not parallel, but BC and DA are parallel because they have the same direction vector. However, AB and CD are not equal in length, but BC and DA are equal in length.
Thus, the quadrilateral ABCD is a trapezium (since one pair of opposite sides, BC and DA, are parallel) but not a parallelogram (since opposite sides AB and CD are not parallel and opposite sides BC and DA are not equal in length).
