Question

Match the formula to its use in proving properties of quadrilaterals

midpoint formula
distance formula
slope formula (if opposite reciprocals)
slope formula (if they are equal)
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Answer 1

The midpoint formula
`\( M = \left( (x_1 + x_2)/(2), (y_1 + y_2)/(2) \right) \)` finds segment centers for parallel lines, the distance formula calculates lengths, and slope formulas (
`\( m = -(x_2 - x_1)/(y_2 - y_1) \)` for perpendicular,
`\( m = (y_2 - y_1)/(x_2 - x_1) \)` for parallel) prove relationships between lines, using coordinates
`\((x_1, y_1)\)` and
`\((x_2, y_2)\)`.

1. Midpoint Formula:

`\[ M = \left( (x_1 + x_2)/(2), (y_1 + y_2)/(2) \right) \]`

- Use: Finding midpoints of segments to prove parallel lines or midpoint relationships.

- Coordinates: Endpoints of the segment:
`\((x_1, y_1)\)` and
`\((x_2, y_2)\)`.

2. Distance Formula:

`\[ D = √((x_2 - x_1)^2 + (y_2 - y_1)^2) \]`

- **Use:** Proving congruent segments, showing sides are equal in length. Proving perpendicular lines by showing a right triangle with legs of equal length.

- Coordinates: Two points:
`\((x_1, y_1)\) and \((x_2, y_2)\)`.

3. Slope Formula (Reciprocal):

`\[ m = -(x_2 - x_1)/(y_2 - y_1) \]`

- Use: Proving perpendicular lines.

- Coordinates: Two points on each line.

4. Slope Formula (Equal):

`\[ m = (y_2 - y_1)/(x_2 - x_1) \]`

- Use: Proving parallel lines.

- Coordinates: Two points on each line.

In all cases:

-
`\(x_1\) and \(y_1\)`: Coordinates of the first point.

-
`\(x_2\)` and
`\(y_2\)`: Coordinates of the second point.