Math Problem Statement
Quadrilateral ABCD is shown on the coordinate plane.
A figure shows a projector is placed 10 meters from a screen ABC with an angle of A 36 degrees projector and AB
C with an angle of A
36 degrees.
What needs to be proven to conclude that quadrilateral ABCD is a parallelogram?
A. side length of AB = side length of CD and slope of BC × slope of CD = -1 B. side length of BC = side length of DA and slope of DA × slope of AB = -1 C. slope of AB = slope of CD and slope of BC = slope of DA D. slope of AB = slope of BC and slope of CD = slope of DA
Solution
To prove that quadrilateral ABCD is a parallelogram, we need to establish that both pairs of opposite sides are parallel. This is one of the defining properties of parallelograms.
- Two lines are parallel if they have the same slope.
- Thus, for quadrilateral ABCD to be a parallelogram, we need to show that the opposite sides, AB and CD, and BC and DA, are parallel.
This means:
- The slope of AB must be equal to the slope of CD.
- The slope of BC must be equal to the slope of DA.
Therefore, the correct answer is:
C.
slope of AB = slope of CD and slope of BC = slope of DA
This condition proves that both pairs of opposite sides are parallel, which is sufficient to conclude that ABCD is a parallelogram.
Would you like more details or have any questions?
Here are 5 related questions to explore this further:
- What other properties can define a parallelogram?
- How can the slopes of diagonals help in proving a quadrilateral is a parallelogram?
- What if only one pair of opposite sides has equal slopes? What type of quadrilateral would that be?
- How can the distance formula be used to prove sides are equal in a parallelogram?
- What are the conditions to prove a quadrilateral is a rectangle or rhombus?
Tip: To prove lines are parallel, always check their slopes—if they are equal, the lines are parallel.
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Math Problem Analysis
Mathematical Concepts
Coordinate Geometry
Properties of Parallelograms
Slopes of Lines
Formulas
Slope formula: (y2 - y1) / (x2 - x1)
Properties of parallelograms: Opposite sides are parallel and equal in length
Theorems
Slope Theorem
Parallel Line Theorem for Parallelograms
Suitable Grade Level
Grades 9-10
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