How to Understand Congruence through Rigid Motion Transformations

Step-by-step Guide: Congruence and Rigid Motions

Defining Congruence:
Geometric figures are deemed congruent if they have the exact same size and shape. Essentially, they would match up perfectly if superimposed.

Exploring Rigid Motions:
These are transformations, also known as isometries, which retain the size or shape of a figure. They include:

• Translations: Shifts that move a shape without rotating or flipping it.
• Rotations: Spinning a shape around a fixed point.
• Reflections: Creating a mirror image by flipping a shape over a line.

1. Linking Congruence with Rigid Motions:
   Shapes are congruent if one can be transformed into the other solely using rigid motions.
2. Recognizing Congruent Figures:
   Search for shapes with matching sides and angles. Additionally, verify if one shape can be remapped to the other using only rigid motions.

Examples

Example 1:
Quadrilateral \( ABCD \) with vertices \( A(1,2) \), \( B(3,2) \), \( C(3,4) \), and \( D(1,4) \)
Quadrilateral \( WXYZ \) with vertices \( W(5,6) \), \( X(7,6) \), \( Y(7,8) \), and \( Z(5,8) \)

Question:
Are the two quadrilaterals congruent through a rigid motion on the coordinate plane?

Solution:
By observing, we can see that each vertex of Quadrilateral \( WXYZ \) is \( 4 \) units to the right and \( 4 \) units up from the corresponding vertex of Quadrilateral \( ABCD \). Thus, a translation (a rigid motion) of \( 4 \) units right and \( 4 \) units up would map \( ABCD \) onto \( WXYZ \).
Hence, the two quadrilaterals are congruent through the rigid motion of translation.

Example 2:

Triangle \( PQR \) with vertices \( P(1,1) \), \( Q(3,1) \), and \( R(2,3) \)
Triangle \( STU \) with vertices \( S(1,3) \), \( T(1,1) \), and \( U(3,2) \)

Question:
Are the two triangles congruent through a rigid motion on the coordinate plane?

Solution:
Triangle \( STU \) seems to be a rotation of Triangle \( PQR \) about some point. If we take the midpoint of \( PQ \), which is \( M(2,1) \), as the center of rotation and rotate \( PQR \) by \( 90^\circ \) counterclockwise, we get Triangle \( STU \). Thus, a rotation (a rigid motion) of \( 90^\circ \) counterclockwise about point \( M \) maps \( PQR \) onto \( STU \).
Hence, the two triangles are congruent through the rigid motion of rotation.

Practice Questions:

1. Rectangle \( MNPQ \) is reflected over the y-axis to form \( M’N’P’Q’ \). Are \( MNPQ \) and \( M’N’P’Q’ \) congruent? Justify your answer.
2. Circle \( A \) has a radius of \( 5 \) units. When translated \( 6 \) units to the right, it forms Circle \( B \). Are circles \( A \) and \( B \) congruent?

Answers:

1. Yes, \( MNPQ \) and \( M’N’P’Q’ \) are congruent. Reflections, being rigid motions, do not alter the size or shape of figures.
2. Yes, both circles are congruent. Translations, another form of rigid motion, maintain the size and shape. Therefore, both circles have a radius of \( 5 \) units.