To determine if triangles \( \triangle MNO \) and \( \triangle QRS \) are congruent, we can analyze the transformations (translations, rotations, reflections) that can map one triangle onto the other.
Step 1: Plot the vertices of both triangles.
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Triangle \( MNO \):
- \( M(2, 4) \)
- \( N(5, 4) \)
- \( O(4, 1) \)
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Triangle \( QRS \):
- \( Q(5, 4) \)
- \( R(8, 4) \)
- \( S(7, 1) \)
Step 2: Check the side lengths of both triangles.
We calculate the lengths of each side of triangles \( MNO \) and \( QRS \) using the distance formula:
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For \( \triangle MNO \):
- \( MN = \sqrt{(5 - 2)^2 + (4 - 4)^2} = \sqrt{3^2 + 0^2} = 3 \)
- \( NO = \sqrt{(4 - 5)^2 + (1 - 4)^2} = \sqrt{(-1)^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10} \)
- \( OM = \sqrt{(2 - 4)^2 + (4 - 1)^2} = \sqrt{(-2)^2 + (3)^2} = \sqrt{4 + 9} = \sqrt{13} \)
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For \( \triangle QRS \):
- \( QR = \sqrt{(8 - 5)^2 + (4 - 4)^2} = \sqrt{3^2 + 0^2} = 3 \)
- \( RS = \sqrt{(7 - 8)^2 + (1 - 4)^2} = \sqrt{(-1)^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10} \)
- \( SQ = \sqrt{(5 - 7)^2 + (4 - 1)^2} = \sqrt{(-2)^2 + (3)^2} = \sqrt{4 + 9} = \sqrt{13} \)
Step 3: Compare side lengths.
The side lengths for both triangles are:
- \( \triangle MNO \): \( 3, \sqrt{10}, \sqrt{13} \)
- \( \triangle QRS \): \( 3, \sqrt{10}, \sqrt{13} \)
Step 4: Congruence criteria.
Since both triangles have the same side lengths, we can conclude that \( \triangle MNO \) is congruent to \( \triangle QRS \) by the Side-Side-Side (SSS) congruence criterion.
Step 5: Transformation reasoning.
To visualize the transformation, we can see:
- Translation: The triangle \( MNO \) can be translated right by 3 units and kept at the same height to align point \( M(2, 4) \) to point \( Q(5, 4) \).
- No further transformation is needed: Since triangles have the same shape and dimensions after translation, they remain congruent as they coincide.
Thus, the answer is:
Yes, \( \triangle MNO \) and \( \triangle QRS \) are congruent by SSS criterion, and can be related by a translation.