Question 1017250: http://prntscr.com/9xsn9a
The answers are parallelogram, rectangle, square, and rhombus.
The correct answer is rhombus, but I'm confused how?
I plugged the diagonals into the slope formula and my final result when I multiplied the products of the slopes was 0/0 each time.
Z (-4,2)
W(1,5)
X(5,2)
Y(1,-1)
I got 0/9 for 2-2 over 5--4 which results in 0/9.
I got -6/0 for -1-5 over 1-1 which results in -6/0 and then I multiplied 0/9 and -6/0 which gave me 0/0, but for the parallelogram to be a rhombus the product of the slopes has to be -1 and I got 0/0. So, I'm not sure why the correct answer is a rhombus. By the way, I got parallelogram for my answer.
Found 2 solutions by Theo, MathTherapy:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the slope of a horizontal line is 0.
the slope of a vertical line is undefined.
the rule that the lines are perpendicular to each other when the product of the slopes is equal to -1 only works when both slopes are defined.
in the case of a horizontal line and a vertical line, that rule can't be used.
the rule that can still be used is that the slope of a line perpendicular to another line must be a negative reciprocal of the slope of that line.
the slope of the horizontal line is 0.
the negative reciprocal of 0 is -1 / 0 which is undefined.
since the slope of the vertical line is undefined, it must be perpendicular to the horizontal line.
here's a proof that the parallelogram is a rhombus if the diagonals are perpendicular.
http://www.algebra.com/algebra/homework/Geometry-proofs/Geometry_proofs.faq.question.392003.html
they did not say, however, that the figure in the graph was a parallelogram.
the proof depends on the figure being a parallelogram.
you are just shown the graph.
the properties of a rhombus are:
the diagonals are perpendicular.
the figure is a parallelogram.
to show the figure is a parallelogram, you have to show that the opposite sides are parallel and are congruent.
once you've shown that, you can then say that it must be a rhombus because, on top of that, the diagonals are perpendicular.
but, if you showed the opposite sides are parallel and that all the sides are congruent, then you've satisfied the definition of a rhombus.
also, in a parallelogram, the diagonals bisect each other.
that is not necessarily true in a quadrilateral, unless the quadrilateral is a parallelogram.
so, besides the fact that you struggled with the diagonals being perpendicular to each other, i think you have to show that the opposite sides are parallel (use their slopes), and that the opposite sides are congruent.
that says it's a parallelogram.
then, either all 4 sides are congruent to each other, which you would probably find after you measured all their lengths.
once you've done that, you don't need to state that the diagonals are perpendicular, although that would be icing on the cake.
before you can prove it's a rhombus, you have to prove it's a parallelogram, because a rhombus is a special kind of parallelogram.
here's a definition of a rhombus from wikipedia at https://en.wikipedia.org/wiki/Rhombus
A simple (non self-intersecting) quadrilateral is a rhombus if and only if it is any one of the following:[6][7]
a quadrilateral with four sides of equal length (by definition)
a quadrilateral in which the diagonals are perpendicular and bisect each other
a quadrilateral in which each diagonal bisects two opposite interior angles
a parallelogram in which a diagonal bisects an interior angle
a parallelogram in which at least two consecutive sides are equal in length
a parallelogram in which the diagonals are perpendicular (an orthodiagonal parallelogram)
any one of these definition defines a rhombus.
Answer by MathTherapy(10555)  (Show Source):
You can put this solution on YOUR website! http://prntscr.com/9xsn9a
The answers are parallelogram, rectangle, square, and rhombus.
The correct answer is rhombus, but I'm confused how?
I plugged the diagonals into the slope formula and my final result when I multiplied the products of the slopes was 0/0 each time.
Z (-4,2)
W(1,5)
X(5,2)
Y(1,-1)
I got 0/9 for 2-2 over 5--4 which results in 0/9.
I got -6/0 for -1-5 over 1-1 which results in -6/0 and then I multiplied 0/9 and -6/0 which gave me 0/0, but for the parallelogram to be a rhombus the product of the slopes has to be -1 and I got 0/0. So, I'm not sure why the correct answer is a rhombus. By the way, I got parallelogram for my answer.
First, Z is not (- 4, 2), but (- 3, 2) instead. Maybe this is the reason why you’re not seeing a rhombus.
This is strategy I would use:
As seen, XZ and WY are its diagonals. Now, the length of diagonal XZ is 8 units, since the points' y coordinates (2) are the same, which
means that the diagonal, XZ is a horizontal line that's parallel to the x-axis. Therefore, this diagonal's length is the difference
between its x-coordinates, or 5 - - 3, or 8 units
Likewise, the length of diagonal: WY is 6, since the points' x coordinates are the same, which means that the diagonal, WY is a vertical
line that's parallel to the y-axis. Therefore, this diagonal's length is the difference between its y-coordinates, or 5 - - 1, or 6 units
With the above information, a rectangle and square can be eliminated, since these quadrilaterals' diagonals are congruent. This leaves
2 choices: a parallelogram or a rhombus.
One major difference between the two is that the rhombus' sides are all congruent, while the parallelogram's are not. Taking two adjacent
sides: WX and XY, we find that the length of WX, using the distance formula, is: 5 units, while the length of XY, using the distance
formula is also 5 units. You could also test the other pair of adjacent sides: WZ and ZY, but this isn’t necessary.
Thus, the quad WXYZ is proven to be a  .
|
|
|
