Question

question 1 - 14
triangle abc is located on a coordinate plane. triangle abc is transformed to create triangle abc using the rule indicated below
(x,y)→(y + 6,x + 5)
are triangles abc and abc congruent? why or why not?
yes, they are congruent because the transformation increases the length of each side by the same amount, and the corresponding angles are equal.
no, they are not congruent because the order of the x - coordinate and the y - coordinate are inverted, which changes the orientation of the triangle, thus changing the measures of the angles at the vertices.
yes, they are congruent because the rule that was used in the transformation represents a rigid motion, so the sides of triangles abc and abc are the same lengths.
no, they are not congruent because the sides of triangle abc are 11 units longer than the sides of triangle abc

Explanation:

A rigid motion (also known as an isometry) is a transformation that preserves the shape and size of a figure. The transformation rule \((x,y)\to(y + 6,x+6)\) is a combination of a reflection (swapping \(x\) and \(y\) coordinates) and a translation (adding 6 to both new coordinates). Rigid motions do not change the lengths of the sides of a triangle. If two triangles have their corresponding sides equal in length (because of the rigid - motion transformation), then they are congruent.

• Option 1: The transformation is not just an increase in height. It is a rigid motion. So, this option is incorrect.
• Option 2: Swapping \(x\) and \(y\) coordinates (along with translation) is a rigid motion and does not change the side - lengths. So, this option is incorrect.
• Option 3: Since the transformation \((x,y)\to(y + 6,x + 6)\) is a rigid motion (a composition of reflection and translation), and rigid motions preserve side - lengths. So, triangles \(ABC\) and \(A'B'C'\) are congruent. This option is correct.
• Option 4: The sides of \(A'B'C'\) are not longer. Rigid motions preserve side - lengths. So, this option is incorrect.

Answer:

C. Yes, they are congruent because the rule that was used in the transformation represents a rigid motion, so the sides of triangles \(ABC\) and \(A'B'C'\) are the same lengths.