To determine whether quadrilateral ABCD is a rhombus, we need to analyze the lengths of its sides. We are given the vertices' coordinates: A(0, 0), B(5, 0), C(5, 4), and D(3, 4). We can use the distance formula to calculate the lengths of the sides:
The distance formula between two points [tex]\( (x_1, y_1) \)[/tex] and [tex]\( (x_2, y_2) \)[/tex] is:
[tex]\[ \text{distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
1. Calculate the length of [tex]\( AB \)[/tex] (between points A and B):
[tex]\[ AB = \sqrt{(5 - 0)^2 + (0 - 0)^2} = \sqrt{25 + 0} = \sqrt{25} = 5 \text{ units} \][/tex]
2. Calculate the length of [tex]\( BC \)[/tex] (between points B and C):
[tex]\[ BC = \sqrt{(5 - 5)^2 + (4 - 0)^2} = \sqrt{0 + 16} = \sqrt{16} = 4 \text{ units} \][/tex]
3. Calculate the length of [tex]\( CD \)[/tex] (between points C and D):
[tex]\[ CD = \sqrt{(5 - 3)^2 + (4 - 4)^2} = \sqrt{4 + 0} = \sqrt{4} = 2 \text{ units} \][/tex]
4. Calculate the length of [tex]\( DA \)[/tex] (between points D and A):
[tex]\[ DA = \sqrt{(3 - 0)^2 + (4 - 0)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \text{ units} \][/tex]
For ABCD to be a rhombus, all four sides should be of equal length.
Upon comparing these values, we find:
- [tex]\( AB = 5 \)[/tex] units
- [tex]\( BC = 4 \)[/tex] units
- [tex]\( CD = 2 \)[/tex] units
- [tex]\( DA = 5 \)[/tex] units
Clearly, not all four sides are of equal length, thus ABCD does not form a rhombus.
By examining the possible justifications:
- ABCD is not a rhombus because it is not a parallelogram since sides [tex]\( BC \)[/tex] and [tex]\( DA \)[/tex] are not parallel: Incorrect, no mention of parallelism needed.
- ABCD is not a rhombus because the lengths of [tex]\( AB \)[/tex] and [tex]\( CD \)[/tex] are 5 units but the lengths of [tex]\( BC \)[/tex] and [tex]\( DA \)[/tex] are different: Correct.
- ABCD is a rhombus because it consists of two pairs of parallel opposite sides: Incorrect.
- ABCD is a rhombus because the lengths of all four sides are 5 units: Incorrect.
Thus, the best justification is:
"ABCD is not a rhombus because the lengths of AB and CD are 5 units but the lengths of BC and DA are different."
