Math Problem Statement
Use the distance formula to determine if the figure is a parallelogram. P (-4, 2), Q (6, 4), R (11, -2), S (2, -3)
Solution
To determine if the quadrilateral is a parallelogram, we can use the distance formula to calculate the lengths of its sides and check if the opposite sides are equal. In a parallelogram, opposite sides must be congruent (i.e., equal in length).
The distance formula is:
Where and are the coordinates of two points.
Step 1: Calculate the lengths of the sides
We will calculate the lengths of the four sides: , , , and .
:
Using the coordinates of points and :
:
Using the coordinates of points and :
:
Using the coordinates of points and :
:
Using the coordinates of points and :
Step 2: Compare the lengths of opposite sides
- and (These are not equal).
- and (These are equal).
Since the opposite sides and are not equal, this is not a parallelogram.
Summary:
The quadrilateral does not satisfy the condition of having opposite sides equal, so it is not a parallelogram.
Would you like further details on this or have additional questions? Here are some related questions you might find interesting:
- What other methods can be used to determine if a quadrilateral is a parallelogram?
- How can we use the slope formula to check if opposite sides are parallel in a quadrilateral?
- What is the condition for a quadrilateral to be a rectangle?
- How can we prove that two sides are equal using coordinate geometry?
- Can we use the distance formula for other types of polygons?
Tip: The distance formula is very useful for verifying side lengths and confirming whether the opposite sides of a quadrilateral are equal when checking for parallelograms or other quadrilaterals!
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Math Problem Analysis
Mathematical Concepts
Coordinate Geometry
Distance Formula
Parallelogram Properties
Formulas
Distance Formula: d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Theorems
Parallelogram Properties: Opposite sides are congruent
Suitable Grade Level
Grades 9-11
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