The vertices of a quadrilateral ABCD are A (0, 5), B(9, 2), C (7, –4), and D(–2, –1). Is ABCD a rectangle? Prove using mathematical evidence and then justify your answer.
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the length of AB is √((9-0)^2 + (2-5)^2) = √81+9) = √90
The slope of AB is (2-5)/(9-0) = -1/3
Now check the other sides, and make sure that CD=AB and BC=AD
CD also has slope -1/3
BC and AD have slope 3
To determine if the quadrilateral ABCD is a rectangle, we need to analyze its properties. A rectangle is a quadrilateral with four right angles, meaning all its interior angles are equal to 90 degrees.
Step 1: Calculate the slopes of each pair of adjacent sides, AB, BC, CD, and DA, to see if they are perpendicular. Two lines are perpendicular if and only if the product of their slopes is -1.
The slope of a line is calculated using the formula: slope = (change in y-coordinate) / (change in x-coordinate).
The slope between points A and B: mAB = (2 - 5) / (9 - 0) = -3 / 9 = -1/3
The slope between points B and C: mBC = (-4 - 2) / (7 - 9) = -6 / -2 = 3
The slope between points C and D: mCD = (-1 + 4) / (-2 - 7) = 3 / -9 = -1/3
The slope between points D and A: mDA = (5 + 1) / (0 + 2) = 6 / 2 = 3
Step 2: Check if opposite sides are parallel. Two line segments are parallel if and only if their slopes are equal.
The slopes of opposite sides AB and CD are -1/3 and -1/3, respectively, which are equal.
The slopes of opposite sides BC and DA are 3 and 3, respectively, which are also equal.
Step 3: Determine if adjacent sides are perpendicular. If the product of the slopes of adjacent sides is -1, then the sides are perpendicular.
The product of slopes -1/3 * 3 = -1, indicating that the sides AB and BC are perpendicular.
Similarly, the product of slopes 3 * -1/3 = -1, showing that the sides BC and CD are perpendicular.
Therefore, we have mathematical evidence to conclude that ABCD is a rectangle because all four angles are right angles, and the adjacent sides are both perpendicular and parallel.
Justification: From the calculation of slopes, we find that the slopes of opposite sides are equal, indicating parallelism. Additionally, the product of the slopes of adjacent sides is -1, confirming perpendicularity. These properties satisfy the conditions for a rectangle. Therefore, we can confidently state that ABCD is indeed a rectangle based on the given coordinates.