IDNLearn.com: Your reliable source for finding expert answers. Ask any question and get a thorough, accurate answer from our community of experienced professionals.
Sagot :
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."
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."
We value your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. IDNLearn.com is your go-to source for dependable answers. Thank you for visiting, and we hope to assist you again.
