Math Problem Statement
Solution
To solve this problem, let's determine which statement best proves that the given quadrilateral is a rectangle. Here’s how we proceed:
Step 1: Recall the Properties of a Rectangle
- Opposite sides are parallel (i.e., they have the same slope).
- 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 and from the diagram:
- , , , .
The slope formula is:
Slopes of the sides:
-
Slope of :
-
Slope of :
-
Slope of :
-
Slope of :
Step 3: Analyze the Slopes
- Opposite sides ( and ) have the same slope (), confirming they are parallel.
- Opposite sides ( and ) have the same slope (), confirming they are parallel.
- Adjacent sides ( and ) have slopes that are opposite reciprocals ( and ), 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:
- How do you verify if a quadrilateral is a parallelogram using slopes?
- What is the importance of opposite reciprocals in geometry?
- How can the distance formula be used to confirm that a shape is a rectangle?
- How do properties of slopes help confirm the presence of right angles?
- 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.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
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
Related Recommendation
Proving a Quadrilateral ABCD is a Parallelogram Using Slopes
Prove a Quadrilateral is a Square Using Distance and Slope Formulas
Verify Rectangle with Slope: Points A(3,2), B(13,4), C(12,9), D(2,7)
How to Prove a Rectangle - Methods and Proofs
Proving Parallel Lines and Rectangular Properties in a Quadrilateral