Question

a quadrilateral on a coordinate plane has vertices $q(-21, -8)$, $r(-10, 10)$, $s(5, 5)$, and $t(3, -16)$. tyron uses the following work to make a conclusion about the classification of the quadrilateral. work: slopes of sides for $overline{qt}$: $\frac{-8 - (-16)}{-21 - 3}=\frac{8}{-24}=-\frac{1}{3}$ for $overline{rs}$: $\frac{10 - 5}{-10 - 5}=\frac{5}{-15}=-\frac{1}{3}$ slopes of diagonals for $overline{qs}$: $\frac{-8 - 5}{-21 - 5}=\frac{-13}{-26}=\frac{1}{2}$ for $overline{rt}$: $\frac{10-(-16)}{-10 - 3}=\frac{26}{-13}=-2$ 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:

Step1: Recall properties of a rhombus

A rhombus is a parallelogram with perpendicular diagonals. To prove a quadrilateral is a rhombus, first we need to show it is a parallelogram (opposite - sides are parallel, i.e., have equal slopes) and then that its diagonals are perpendicular (product of diagonal - slopes is - 1).

Step2: Analyze the given work

The work only calculates the slopes of the sides and diagonals. Calculating the slopes of the sides can help check for parallel - sides (to prove it's a parallelogram), but the given work doesn't use the side - slope information to show that opposite sides are parallel. Just calculating the slopes of the diagonals is not enough to conclude it's a rhombus without first showing it's 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.