Math Problem Statement

show that the points (0,5) (3,-4) (8,0) and (3,-1) are the vertices of a parallelogram

Solution

To show that the points (0,5)(0, 5), (3,4)(3, -4), (8,0)(8, 0), and (3,1)(3, -1) are the vertices of a parallelogram, we need to verify that either:

  1. Opposite sides are parallel, or
  2. Diagonals bisect each other.

Step 1: Label the points

Let the points be labeled as:

  • A(0,5)A(0, 5)
  • B(3,4)B(3, -4)
  • C(8,0)C(8, 0)
  • D(3,1)D(3, -1)

Step 2: Check if opposite sides are parallel

To determine if opposite sides are parallel, we need to calculate the slopes of opposite sides.

Slope of line ABAB:

The slope of the line passing through two points (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is given by:

slope of AB=y2y1x2x1=4530=93=3\text{slope of } AB = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-4 - 5}{3 - 0} = \frac{-9}{3} = -3

Slope of line CDCD:

slope of CD=1038=15=15\text{slope of } CD = \frac{-1 - 0}{3 - 8} = \frac{-1}{-5} = \frac{1}{5}

So, lines ABAB and CDCD are not parallel.

Slope of line BCBC:

slope of BC=0(4)83=45\text{slope of } BC = \frac{0 - (-4)}{8 - 3} = \frac{4}{5}

Slope of line ADAD:

slope of AD=1530=63=2\text{slope of } AD = \frac{-1 - 5}{3 - 0} = \frac{-6}{3} = -2

So, lines BCBC and ADAD are also not parallel.

Step 3: Check if diagonals bisect each other

The diagonals of a parallelogram bisect each other, meaning their midpoints should be the same.

Midpoint of diagonal ACAC:

The midpoint of a line segment with endpoints (x1,y1)(x_1, y_1) and (x2,y2)(x_2, y_2) is given by:

Midpoint of AC=(0+82,5+02)=(4,2.5)\text{Midpoint of } AC = \left(\frac{0 + 8}{2}, \frac{5 + 0}{2}\right) = (4, 2.5)

Midpoint of diagonal BDBD:

Midpoint of BD=(3+32,4+(1)2)=(3,2.5)\text{Midpoint of } BD = \left(\frac{3 + 3}{2}, \frac{-4 + (-1)}{2}\right) = (3, -2.5)

Since the midpoints of diagonals ACAC and BDBD are not the same, the diagonals do not bisect each other.

Conclusion:

The points do not form a parallelogram because neither the opposite sides are parallel nor do the diagonals bisect each other.


Would you like further details or clarifications?

Here are five related questions for further exploration:

  1. What conditions must be satisfied for points to form a rectangle?
  2. How can we verify if four points form a rhombus?
  3. What is the formula for calculating the area of a parallelogram given its vertices?
  4. How do the slopes of diagonals in a parallelogram relate to its shape?
  5. Can we classify quadrilaterals based on side lengths and angles?

Tip: When working with coordinates, always double-check slope calculations to avoid errors.

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Math Problem Analysis

Mathematical Concepts

Coordinate Geometry
Slopes of Lines
Midpoints
Properties of Parallelograms

Formulas

Slope of a line: (y2 - y1) / (x2 - x1)
Midpoint of a line segment: ((x1 + x2) / 2, (y1 + y2) / 2)

Theorems

If opposite sides of a quadrilateral are parallel, it is a parallelogram.
The diagonals of a parallelogram bisect each other.

Suitable Grade Level

Grades 9-10