Solve Rhombus: Coordinate Geometry Question

In summary, the conversation discusses how to prove that a given quadrilateral is a rhombus. The method suggested is to show that all four sides are equal and that the diagonals are not equal, using either the approach of finding the gradients of the sides or directly showing that the opposite sides are parallel and have equal length. The concept of gradient or slope of a line is also briefly explained.
  • #1
denian
641
0
just want to ask a simple question

a quadrilateral has the vertices A ( 1,4 ), B (9,5 ) , C ( 5,-2) and D (-3,-3). show that the quadrilateral is a rhombus

what i do is find gradient AB, CD, AD and BC.
and then state which and which have the same gradient... and hence the opposite sides are parallel.

is that all i should do?
 
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  • #2
Unless they're doing something tricky, the sides should be AB, BC, DC, and AD. Show that AB = DC, BC = AD, and |AB| = |BC|, and you're done. Basically a rhombus as two defining properties:

1) all four sides have the same length
2) opposite sides are parallel

In truth, it's sufficient to simply show that all four sides have the same length, and property 2 will follow, so that's one alternative approach. I would say try both, and see which one feels easier/more efficient. As for your approach, I don't know what you mean by gradient, so I can't say whether it will work or not.
 
  • #3
The two sufficient conditions to prove are:

1. All 4 sides are equal (opposite sides are parallel).
2. The diagonals are not equal.

AKG: By gradient, denian means slope of a line joining [tex](x_{1}, y_{1})[/tex] and [tex](x_{2}, y_{2})[/tex] defined as:

[tex]
m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}
[/tex]

Cheers
Vivek
 
  • #4
thank you!
 

1. What is a rhombus in coordinate geometry?

A rhombus is a four-sided polygon with all sides equal in length. In coordinate geometry, it is represented by four points on a coordinate plane, where the opposite sides are parallel to each other and the diagonals bisect each other at right angles.

2. How do you find the area of a rhombus in coordinate geometry?

To find the area of a rhombus in coordinate geometry, you can use the formula A = (1/2) x d1 x d2, where d1 and d2 are the lengths of the diagonals. You can also use the formula A = (1/2) x base x height, where the base and height are the lengths of any two adjacent sides.

3. How do you find the perimeter of a rhombus in coordinate geometry?

The perimeter of a rhombus in coordinate geometry can be found by adding the lengths of all four sides. You can also use the formula P = 4 x side length.

4. How do you prove that a quadrilateral on a coordinate plane is a rhombus?

To prove that a quadrilateral on a coordinate plane is a rhombus, you can use the properties of a rhombus, such as equal side lengths and perpendicular diagonals. You can also use the distance formula to calculate the length of all four sides and the midpoint formula to show that the diagonals bisect each other at right angles.

5. How can you use the coordinates of a rhombus to find its properties?

The coordinates of a rhombus can be used to find its properties such as side lengths, angles, area, and perimeter. By calculating the distance between the points and using the slope formula, you can determine if the sides are equal in length and if the diagonals are perpendicular. You can also use the Pythagorean theorem to find the length of the diagonals and determine if the rhombus is a right rhombus.

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