Question 265609: The vertices of quadrilateral ABCD are A(4,0), B(13,3), C(12,6) and D(3,3).
How do I show ABCD is a rectangle? (I've already proved it to be a quadrilateral.)
Found 2 solutions by dabanfield, solver91311:
Answer by dabanfield(803)   (Show Source):
You can put this solution on YOUR website! Show that the lines AB and CD are parallel (same slopes) and that the lines BC and AD are also parallel (same slopes). Then show that the line AB is perpendicular to BC (slopes are negative reciprocals) and that AD is perpendiucular to CD (slopes are negative reciprocals).
Then show that the distance from A to B is equal to the distance from C to D and that the distance from BC is equal to the distance from AD.
Answer by solver91311(24713)  (Show Source):
You can put this solution on YOUR website!
Prove that the line that contains segment AB is perpendicular to the line that contains segment BC. 2nd prove that the line containing segment AB is parallel to the line containing segment CD. And then prove that the line that contains the segment CD is perpendicular to the line that contains the segment DA.
A line contains a given segment if both of the endpoints of the segment are contained in the line. Two lines are perpendicular if their slopes are negative reciprocals, that is:
And two lines are parallel if their slopes are equal, that is:
So, just calculate the slope of the line that contains the points AB using the slope formula:
where ) and ) are the coordinates of points A and B.
Then do the same thing for points B and C.
Finally, determine if the slope of the line through AB is the negative reciprocal of the line through BC.
If not, you can quit right here because the quadrilateral is definitely NOT a rectangle. If they are negative reciprocals, then calculate the slope of CD. If it is not equal to the slope of AB, quit -- again, not a rectangle. If the slopes of AB and CD are equal, calculate the slope of DA. If the slope of the line containing DA is the negative reciprocal of the slope of the line containing CD, you have a rectangle, otherwise not.
John
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