Question

sammy is classifying quadrilateral efgh. how can she use coordinate geometry to determine whether efgh is a parallelogram? e(0,4), f(0,7), g(1,5), h(1,2) sammy can use the distance formula to show that the opposite sides have equal slopes. sammy can use the distance formula to show that all sides are equal. sammy can use the slope formula to show that the opposite sides have equal slopes. sammy can use the slope formula to show that all sides are equal.

Explanation:

Step1: Recall the property of a parallelogram

A parallelogram has opposite sides that are parallel. In coordinate geometry, parallel lines have equal slopes.

Step2: Analyze the options

• Option 1: Using the slope formula to show all sides are equal is incorrect. Equal - length sides do not necessarily mean it is a parallelogram (e.g., a rhombus has equal - length sides but needs more properties to be a parallelogram).
• Option 2: Using the slope formula to show that opposite sides have equal slopes. If \(m_{EF}=m_{GH}\) and \(m_{FG}=m_{EH}\), then \(EF\parallel GH\) and \(FG\parallel EH\), which satisfies the definition of a parallelogram.
• Option 3: Using the distance formula to show all sides are equal is for a rhombus (a special type of parallelogram, but not the general way to check for a parallelogram).
• Option 4: Using the distance formula to show that opposite sides have equal slopes is a wrong approach as the distance formula gives the length of a line segment, not the slope.

Answer:

B. Sammy can use the slope formula to show that the opposite sides have equal slopes.