Question

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what is the distance between point c and point d?
2 units
which expression represents the distance between point c and point d?

-4	+	-2		3	+	-2
-4	-	-2		3	-	-4

graph: coordinate plane with x-axis from -5 to 5, y-axis from -5 to 5. point c at (3, -2), point d at (3, -4)

Explanation:

Step1: Analyze coordinates of C and D

Point \( C(3, -2) \), Point \( D(3, -4) \). Since x - coordinates are same, distance is vertical. The y - coordinates are \(-2\) and \(-4\). The distance between two points with same x - coordinate is \(|y_1 - y_2|\). Here, \( y_1=-2 \), \( y_2 = - 4 \), so \(|-2-(-4)|=|-2 + 4|=|2| = 2\), or using absolute values of y - coordinates: \(|-4|-|-2|\) (since \(-4\) is more negative, the distance is \(|-4|-|-2|=4 - 2=2\)). Also, check the expressions:

• \(|-4|+|-2| = 4 + 2=6\) (wrong)
• \(|3|+|-2|=3 + 2 = 5\) (wrong)
• \(|-4|-|-2|=4 - 2 = 2\) (correct)
• \(|3|-|-4|=3 - 4=-1\) (absolute value makes it wrong as distance can't be negative, and magnitude is 1, wrong)

Step2: Confirm the correct expression

The vertical distance between \( C(3,-2) \) and \( D(3,-4) \) is calculated by the difference of absolute values of y - coordinates (since they are on the same vertical line \( x = 3 \)). The y - coordinates are \(-2\) and \(-4\), their absolute values are \(|-2| = 2\) and \(|-4|=4\). The distance is \(|-4|-|-2|\) as \(4-2 = 2\), which matches the distance between the points.

Answer:

\(|-4|-|-2|\)