Math Problem Statement

Which of the following statements could be used to prove ABCD is a rectangle?

Solution

To solve this problem, let's determine which statement best proves that the given quadrilateral ABCDABCD is a rectangle. Here’s how we proceed:

Step 1: Recall the Properties of a Rectangle

  1. Opposite sides are parallel (i.e., they have the same slope).
  2. Adjacent sides are perpendicular (i.e., their slopes are opposite reciprocals).

Step 2: Calculate the Slopes of Each Side

Using the coordinates of the vertices A,B,C,A, B, C, and DD from the diagram:

  • A(1,2)A(1, 2), B(3,4)B(3, 4), C(5,3)C(5, 3), D(3,1)D(3, 1).

The slope formula is: m=y2y1x2x1.m = \frac{y_2 - y_1}{x_2 - x_1}.

Slopes of the sides:

  1. Slope of ABAB: mAB=4231=22=1.m_{AB} = \frac{4 - 2}{3 - 1} = \frac{2}{2} = 1.

  2. Slope of BCBC: mBC=3453=12.m_{BC} = \frac{3 - 4}{5 - 3} = \frac{-1}{2}.

  3. Slope of CDCD: mCD=1335=22=1.m_{CD} = \frac{1 - 3}{3 - 5} = \frac{-2}{-2} = 1.

  4. Slope of DADA: mDA=2113=12=12.m_{DA} = \frac{2 - 1}{1 - 3} = \frac{1}{-2} = -\frac{1}{2}.

Step 3: Analyze the Slopes

  • Opposite sides (ABAB and CDCD) have the same slope (m=1m = 1), confirming they are parallel.
  • Opposite sides (BCBC and DADA) have the same slope (m=12m = -\frac{1}{2}), confirming they are parallel.
  • Adjacent sides (ABAB and BCBC) have slopes that are opposite reciprocals (11 and 12-\frac{1}{2}), confirming perpendicularity.

Step 4: Select the Correct Statement

The condition for a rectangle is adjacent sides being perpendicular, which is shown by their slopes being opposite reciprocals.

The correct answer is:

The slopes are opposite reciprocals for adjacent sides.


Do you want further clarification or detailed steps on the slope calculation? Here are some related questions to expand on this:

  1. How do you verify if a quadrilateral is a parallelogram using slopes?
  2. What is the importance of opposite reciprocals in geometry?
  3. How can the distance formula be used to confirm that a shape is a rectangle?
  4. How do properties of slopes help confirm the presence of right angles?
  5. What are other methods (besides slopes) to prove a quadrilateral is a rectangle?

Tip: Always double-check the coordinates and calculations to avoid small errors, especially when analyzing slopes.

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Math Problem Analysis

Mathematical Concepts

Analytic Geometry
Properties of Rectangles
Slopes of Lines

Formulas

Slope formula: m = (y2 - y1) / (x2 - x1)

Theorems

Properties of Rectangles: Opposite sides are parallel; adjacent sides are perpendicular

Suitable Grade Level

Grades 8-10