For coordinates A(1,-4) B(5,-1) C(8,-5) D(4,-8)
Prove ABCD is a rectangle through the use of a coordinate paragraph.
Please I have struggled and i am unable to find the answer
I would first of all plot the points. That would give you an idea which sides would form the rectangle.
Since in a rectangle:
1. opposite sides are parallel, you can show this by finding the slopes
2. slopes of adjacent sides are negative reciprocals of each other. Use your sketch to decide which pairs of slopes to compare.
Let me know how you made out.
AB=3/4
BC=-4/3
CD=3/4
DA=-4/3
Paragraph
Beginning-In Order to Prove that the rectangle ABCD is a quadrilateral, there must be four right angles.
Middle-AB=3/4 BC=-4/3 CD=3/4 DA=-4/3 AB=CD BC=DA
End-Since each of the two segments are an opposite reciprocal to each other, the rectangle ABCD is a quadrilateral.
Good job, you have
1. shown that opposite sides are parallel
2. adjacent sides form a right angle.
Thank You, I will try to do well on my test tomorrow.
To prove that ABCD is a rectangle using a coordinate paragraph, we need to show that its sides are perpendicular to each other and that its diagonals are congruent. We can do this by calculating the slopes of the sides and the lengths of the diagonals.
Let's start by finding the slopes of the sides AB, BC, CD, and DA. The slope of a line passing through two points (x1, y1) and (x2, y2) is given by the formula:
Slope = (y2 - y1) / (x2 - x1)
Using this formula, we can calculate the slopes as follows:
Slope of AB:
Given A(1, -4) and B(5, -1)
Slope_AB = (-1 - (-4)) / (5 - 1) = -3 / 4
Slope of BC:
Given B(5, -1) and C(8, -5)
Slope_BC = (-5 - (-1)) / (8 - 5) = -4 / 3
Slope of CD:
Given C(8, -5) and D(4, -8)
Slope_CD = (-8 - (-5)) / (4 - 8) = -3 / 4
Slope of DA:
Given D(4, -8) and A(1, -4)
Slope_DA = (-4 - (-8)) / (1 - 4) = 4 / 3
Now that we have calculated the slopes, let's check if the opposite sides of the quadrilateral are perpendicular. If two lines are perpendicular, the product of their slopes must be -1.
Product of slopes of AB and CD:
Slope_AB * Slope_CD = (-3 / 4) * (-3 / 4) = 9 / 16 ≠ -1
So, AB and CD are not perpendicular.
Product of slopes of BC and DA:
Slope_BC * Slope_DA = (-4 / 3) * (4 / 3) = -16 / 9 ≠ -1
So, BC and DA are not perpendicular either.
Since the opposite sides are not perpendicular, we have proved that ABCD is not a rectangle.
Please note that a rectangle must have perpendicular sides, but in this case, the sides AB and CD, as well as BC and DA, are not perpendicular. Therefore, ABCD cannot be considered a rectangle.