Question

How do I prove congruence between two geometric figures using the coordinate plane?

Answer 1

To prove congruence between two geometric figures using a coordinate plane, calculate the distances, check for parallelism, and ensure corresponding angles are equivalent by using distance and slope formulas and verifying invariance under rotation.

Step-by-step explanation:

To prove congruence between two geometric figures using the coordinate plane, one would generally follow these steps:

1. Identify the coordinates of the corresponding vertices of the two figures.
2. Use the distance formula, d = \(√((x_2-x_1)^2 + (y_2-y_1)^2)\), to find the distances between corresponding vertices. This is essential to determine if the sides are congruent.
3. Calculate slopes of the corresponding sides to check for parallelism, using the slope formula, m = (y_2-y_1)/(x_2-x_1).
4. Check for angles' congruence by verifying if the products of the slopes of the four pairs of sides are -1 (this implies that the angles are right angles), or by using the direction angle of the line or vector formed by the two points.
5. If all corresponding sides are of equal length and all corresponding angles are equal, then the two figures are congruent.

In special cases, such as when the figures have been rotated, you could show that the distance between two points is invariant under rotations by verifying that the Pythagorean Theorem holds in the rotated coordinate system.