2. Prove that the quadrilateral A(-3, 2), B(-2, 6), C(2, 7), and D(1, 3) is a rhombus.
1.Prove that the diagonals of a rectangle A(0, 0), B(0, 4), C(7, 4), and D(7,0) are congruent.
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1. 2 distance and conclusion
2.4 distance 4 slope and conclusion
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Answer:
Proving that the quadrilateral A(-3, 2), B(-2, 6), C(2, 7), and D(1, 3) is a rhombus:
A rhombus is a quadrilateral with all sides congruent. Additionally, a rhombus has opposite angles congruent and its diagonals are perpendicular bisectors of each other.
To prove that ABCD is a rhombus, we need to show that all four sides are congruent and opposite angles are congruent, and that its diagonals are perpendicular bisectors of each other.
To show that all four sides are congruent:
We can calculate the distance between each pair of adjacent vertices and see if they are equal.
AB = √[(-2 + 3)^2 + (6 - 2)^2] = √10
BC = √[(2 + 2)^2 + (7 - 6)^2] = √5
CD = √[(1 - 2)^2 + (3 - 7)^2] = √20
DA = √[(-3 - 1)^2 + (2 - 3)^2] = √20
Since all four distances are not equal, ABCD is not a rhombus.
Therefore, the given quadrilateral is not a rhombus.
Proving that the diagonals of a rectangle A(0, 0), B(0, 4), C(7, 4), and D(7,0) are congruent:
A rectangle is a parallelogram with four right angles. The diagonals of a rectangle are congruent.
To prove that AC and BD are congruent, we can calculate their lengths and show that they are equal.
AC = √[(7 - 0)^2 + (4 - 0)^2] = √65
BD = √[(7 - 0)^2 + (0 - 4)^2] = √65
Since both diagonals have the same length, AC and BD are congruent.
Therefore, we have proven that the diagonals of the given rectangle are congruent.
Finding distances and slopes for A(0, 0), B(0, 4), C(7, 4), and D(7,0):
Distance between A and B:
AB = √[(0 - 0)^2 + (4 - 0)^2] = 4
Distance between B and C:
BC = √[(7 - 0)^2 + (4 - 4)^2] = 7
Distance between C and D:
CD = √[(7 - 7)^2 + (0 - 4)^2] = 4
Distance between D and A:
DA = √[(0 - 7)^2 + (0 - 0)^2] = 7
Slope of AB:
mAB = (4 - 0)/(0 - 0) = undefined (since the denominator is zero)
Slope of BC:
mBC = (4 - 4)/(7 - 0) = 0
Slope of CD:
mCD = (0 - 4)/(7 - 7) = undefined (since the denominator is zero)
Slope of DA:
mDA = (0 - 0)/(0 - 7) = 0
Conclusion:
We have shown that the distance between the diagonals AC and BD of the given rectangle are equal, and the distances between the vertices are 4, 7, 4, and 7. Additionally, we have found that the slopes