Question

tyron used the following work to make a conclusion about the classification of the quadrilateral
work:
slopes of sides
for \\( \overline { q t } \\) :
\\( \frac { - 4 - ( - 10 ) } { 13 - 3 } = \frac { 6 } { 10 } = \frac { 3 } { 5 } \\)
for \\( \overline { r s } \\) :
\\( \frac { 10 - 4 } { 10 - 5 } = \frac { 6 } { 5 } \\)
slopes of diagonals
for \\( \overline { q s } \\) :
\\( \frac { - 4 - 4 } { 13 - 5 } = \frac { - 8 } { 8 } = - 1 \\)
for \\( \overline { r t } \\) :
\\( \frac { 10 - ( - 10 ) } { 10 - 3 } = \frac { 20 } { 7 } \\)
conclusion: the quadrilateral is a rhombus.
which of these best describes tyrons work and conclusion?
the work shown is correct. the conclusion is correct because a parallelogram with perpendicular diagonals is a rhombus.
the work shown is correct. the conclusion is incorrect because the work does not prove that the quadrilateral is even a parallelogram, let alone a rhombus.
the work shown is incorrect because the calculated slopes are all reciprocals of the correct slopes. the conclusion is correct because a parallelogram with perpendicular diagonals is a rhombus.
the work shown is incorrect because the calculated slopes are all reciprocals of the correct slopes. the conclusion is incorrect because the work does not prove that the quadrilateral is even a parallelogram, let alone a rhombus.

Explanation:

To determine if a quadrilateral is a rhombus, we first need to establish that it is a parallelogram (opposite sides parallel). The work only calculates slopes of sides and diagonals but does not show that opposite sides are parallel (no comparison of slopes of \( \overline{QT}\) and \( \overline{RS}\) with slopes of other - pair of opposite sides). Also, for a parallelogram with perpendicular diagonals to be a rhombus, the quadrilateral must first be a parallelogram.

Answer:

The work shown is correct. The conclusion is incorrect because the work does not prove that the quadrilateral is even a parallelogram, let alone a rhombus.