Determine the most precise name for KLMN (parallelogram, rhombus, rectangle, or square) given K(-5,-1), L(-2, 4), M(3, 1), N(0,-4). Explain how you determined your answer. You must support your answer using length and/or slope.
Answer (2)
Calculate the lengths of all sides of the quadrilateral using the distance formula and determine that all sides are equal.
Calculate the slopes of all sides using the slope formula and verify that opposite sides have equal slopes, confirming it's a parallelogram.
Check if adjacent sides are perpendicular by multiplying their slopes; if the product is -1, they are perpendicular, indicating a rectangle.
Conclude that since all sides are equal and adjacent sides are perpendicular, the quadrilateral is a square .
Explanation
Analyze the given data First, let's analyze the given coordinates of the quadrilateral KLMN: K(-5, -1), L(-2, 4), M(3, 1), and N(0, -4). Our goal is to determine the most precise classification for this quadrilateral: parallelogram, rhombus, rectangle, or square. To do this, we'll calculate the lengths and slopes of its sides.
Calculate the lengths of the sides Next, we calculate the lengths of the sides using the distance formula: d = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 .
Length of KL: K L = ( − 2 − ( − 5 ) ) 2 + ( 4 − ( − 1 ) ) 2 = ( 3 ) 2 + ( 5 ) 2 = 9 + 25 = 34 ≈ 5.83
Length of LM: L M = ( 3 − ( − 2 ) ) 2 + ( 1 − 4 ) 2 = ( 5 ) 2 + ( − 3 ) 2 = 25 + 9 = 34 ≈ 5.83
Length of MN: MN = ( 0 − 3 ) 2 + ( − 4 − 1 ) 2 = ( − 3 ) 2 + ( − 5 ) 2 = 9 + 25 = 34 ≈ 5.83
Length of NK: N K = ( − 5 − 0 ) 2 + ( − 1 − ( − 4 ) ) 2 = ( − 5 ) 2 + ( 3 ) 2 = 25 + 9 = 34 ≈ 5.83
Since all sides have equal lengths ( K L = L M = MN = N K = 34 ), KLMN is either a rhombus or a square.
Calculate the slopes of the sides Now, let's calculate the slopes of the sides using the slope formula: m = x 2 − x 1 y 2 − y 1 .
Slope of KL: m K L = − 2 − ( − 5 ) 4 − ( − 1 ) = 3 5 ≈ 1.67
Slope of LM: m L M = 3 − ( − 2 ) 1 − 4 = 5 − 3 = − 0.6
Slope of MN: m MN = 0 − 3 − 4 − 1 = − 3 − 5 = 3 5 ≈ 1.67
Slope of NK: m N K = − 5 − 0 − 1 − ( − 4 ) = − 5 3 = − 0.6
Since the slopes of opposite sides are equal ( m K L = m MN and m L M = m N K ), the opposite sides are parallel. This confirms that KLMN is a parallelogram.
Check for perpendicularity of adjacent sides To determine if KLMN is a rectangle or a square, we need to check if adjacent sides are perpendicular. We can do this by checking if the product of their slopes is -1.
Check if KL and LM are perpendicular: m K L ⋅ m L M = 3 5 ⋅ 5 − 3 = − 1 . Since the product of the slopes is -1, KL and LM are perpendicular.
Since adjacent sides are perpendicular, KLMN is a rectangle. Also, since all sides are equal, KLMN is a square.
Conclusion Since KLMN has all sides of equal length and adjacent sides are perpendicular, KLMN is a square. Therefore, the most precise name for KLMN is a square.
Final Answer The most precise name for KLMN is square.
Examples
In architecture, determining the precise shape of quadrilaterals is crucial for designing structures with specific properties. For example, knowing if a foundation is perfectly square ensures equal weight distribution and structural stability. Similarly, in computer graphics, accurately classifying quadrilaterals is essential for rendering images and creating realistic 3D models. This problem demonstrates how calculating lengths and slopes can be applied to real-world scenarios requiring precise geometric shapes.
The quadrilateral KLMN is classified as a square because all four sides are of equal length and all adjacent sides are perpendicular to each other.
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