The lengths of each side of the triangle LMN are, LM = 4 units, MN = 3.606 units, and NL = 2.236 units.
The lengths of each side of the triangle L'M'N' are, L'M' = 4 units, M'N' = 3.606 units, and N'L' = 2.236 units.
We can conclude that all corresponding sides are equal, and the two triangles are congruent.
All angles must be congruent, as the image L'M'N' has been parallelly translated from the preimage LMN.
The coordinates of the image are L' (-4, 1), M' (-4, 5), and N' (-2, 2). The length of each side of the triangle LMN:
LM = √((2 - 2)² + (-6 - (-2))²),
or, LM = √((0² + (-4)²),
or, LM = √16 = 4 units.
MN = √((2 - 4)² + (-2 - (-5))²),
or, MN = √((-2)² + (3)²),
or, MN = √(4 + 9) = √13 = 3.606 units.
NL = √((4 - 2)² + (-5 - (-6))²),
or, NL = √((2)² + (1)²)
or, NL = √(4 + 1) = √5 units = 2.236 units.
The length of each side of the triangle L'M'N':
L'M' = √((-4 - (-4))² + (5 - 1)²),
or, L'M' = √((0² + (4)²),
or, L'M' = √16 = 4 units.
M'N' = √((-4 - (-2))² + (5 - 2)²),
or, M'N' = √((-2)² + (3)²),
or, M'N' = √(4 + 9) = √13 = 3.606 units.
N'L' = √((-4 - (-2))² + (1 - 2))²),
or, N'L' = √((2)² + (1)²)
or, N'L' = √(4 + 1) = √5 units = 2.236 units.
Now, since LM = L'M', MN = M'N', and NL = N'L', that, is all the corresponding sides are equal, we can conclude that all corresponding sides are equal, and the two triangles are congruent.
Since the triangles are congruent, the image L'M'N' has been parallelly translated from the preimage LMN.
Thus, all angles must be congruent, as the image L'M'N' has been parallelly translated from the preimage LMN.
Every point moves the same amount and in the same direction throughout a translation. As a result of the picture being identical in size and shape to the preimage, a translation is referred to as a rigid transformation or isometry.
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