in coordinate geometry how can we prove that a quadrilateral is a rhombus and not a square?

Question

if the coordiantes of a quadrilateral TEAM are T(-2,3), E(-5,-4),A(2,-1), AND M(5,6). I hav to prove that the quadrilateral is a rhombus and not a square.

Answer 1

The difference between the rhombus and a square is that the angles of a square are all 90 degrees. So find the gradient of each side.
Gradient of TE = (3-(-4))/((-2)-(-5)) = 7/3
Gradient of EA = ((-1)-(-4))/(2-(-5)) = 3/7
Gradient of AM = (6-(-1))/(5-2) = 7/3
Gradient of MT = (6-3)/(5-(-2)) = 3/7

From the gradients, we know that TEAM cannot be a square as the gradient of any two adjacent sides do not meet the requirements of m1 x m2 = -1 (which proves that the two sides are perpendicular).

Answer 2

TE ,EA, AM and MT all have magnitude
given by √(3² + 7²) = √58
Thus all sides are equal.
Consider dot product of vectors TE and TM
TE is vector (3 - 7)
TM is vector (7 - 3)
Dot product 42
As dot product does not = zero, TE and TM are not at right angles.
Thus figure is a rhombus (all sides equal) but is not a square.

Answer 3

Mathematically, the definition of a square is a four sided two dimensional shape that has all sides equal and all of it's angles are right angles. If you're talking about the expression from the 70's and 80's, it means a person that doesn't fit in. There you go. I'm glad I could help. :)

Answer 4

u must prove that the lines that are formed by those points intersect at right angles, so the slopes of those lines must be negative recipricols of each other. draw the point on a coordinate plane to see wat sides u must prove are prependicular.

Answer 5

squares have matching sides on all sides, rhombuses are slanted.

Answer 6

Quick way is to do it with a graph.