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Chapter 2: Problem 59

Answer the following. Are the points \(A(1,1), B(5,2), C(3,4),\) and \(D(-1,3)\) the vertices of a parallelogram (opposite sides equal in length)? of a rhombus (all sides equal in length)?

Short Answer

Expert verified
The points form a parallelogram but not a rhombus.

Step by step solution

01

- Find the distance between points A and B

Use the distance formula to determine the length of side AB: \[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] For points A(1,1) and B(5,2): \[ AB = \sqrt{(5-1)^2 + (2-1)^2} = \sqrt{16 + 1} = \sqrt{17} \]
02

- Find the distance between points B and C

Use the distance formula to determine the length of side BC: \[ BC = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] For points B(5,2) and C(3,4): \[ BC = \sqrt{(3-5)^2 + (4-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \]
03

- Find the distance between points C and D

Use the distance formula to determine the length of side CD: \[ CD = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] For points C(3,4) and D(-1,3): \[ CD = \sqrt{(3-(-1))^2 + (4-3)^2} = \sqrt{16 + 1} = \sqrt{17} \]
04

- Find the distance between points D and A

Use the distance formula to determine the length of side DA: \[ DA = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] For points D(-1,3) and A(1,1): \[ DA = \sqrt{(1 - (-1))^2 + (1 - 3)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \]
05

- Check for opposite sides' equality

Compare the lengths of opposite sides to see if they are equal: \[ AB = CD = \sqrt{17} \] \[ BC = DA = 2\sqrt{2} \] Since opposite sides are equal, the points form a parallelogram.
06

- Check if all sides are equal

Since the lengths found are: - AB = \( \sqrt{17} \) - BC = \( 2\sqrt{2} \) - CD = \( \sqrt{17} \) - DA = \( 2\sqrt{2} \) Not all sides are equal, so the points do not form a rhombus.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Parallelogram
A parallelogram is a four-sided figure (quadrilateral) with opposite sides that are equal in length and parallel. Imagine a slightly skewed rectangle. If you were to draw two lines between two sets of vertices, those lines (diagonals) would bisect each other.

Using the distance formula, we found that:
  • AB = \( \sqrt{17} \)
  • CD = \( \sqrt{17} \)
  • BC = \( 2\sqrt{2} \)
  • DA = \( 2\sqrt{2} \)
This confirms that the opposite sides of this quadrilateral are equal, thus forming a parallelogram.
Rhombus
A rhombus is another type of quadrilateral, but with all four sides of equal length. It's a special kind of parallelogram. Think of it as a 'pushed over' square.

In this specific exercise, using the distance formula, we observed:
  • AB = \( \sqrt{17} \)
  • BC = \( 2\sqrt{2} \)
  • CD = \( \sqrt{17} \)
  • DA = \( 2\sqrt{2} \)
Since not all sides are equal, the quadrilateral in question isn't a rhombus.
Coordinate Geometry
Coordinate geometry, also known as analytic geometry, involves plotting geometric shapes on the coordinate plane. This technique combines algebra and geometry to solve problems involving distances, midpoints, and slopes of geometric figures.

Here, we used the coordinates of the points A(1,1), B(5,2), C(3,4), and D(-1,3) to apply the distance formula.

Coordinate geometry helps us visually understand shapes and their properties by mapping them out on a grid, which can make abstract concepts more tangible.
Distance Between Points
The distance formula helps us find the straight-line distance between two points in a coordinate plane. It鈥檚 derived from the Pythagorean theorem. The formula is:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Let鈥檚 break it down:
  • \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of two points.
  • Subtract the x-coordinates and y-coordinates respectively.
  • Square each of the differences.
  • Add these squares together.
  • Take the square root of the sum to get the distance.
For instance, in the exercise, to find the distance between points A(1,1) and B(5,2), we calculated: \[ AB = \sqrt{(5-1)^2 + (2-1)^2} = \sqrt{16 + 1} = \sqrt{17} \]

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