To determine the lengths of lines \( \overline{AB} \) and \( \overline{BC} \) in triangle \( ABC \), we first need the coordinates of points \( A \), \( B \), and \( C \). Similarly, for triangle \( PQR \), we need the coordinates of points \( P \), \( Q \), and \( R \).
The lengths of the segments can be found using the distance formula, which is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
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Calculating \( \overline{AB} \): If \( A(x_1, y_1) \) and \( B(x_2, y_2) \) are the coordinates of points \( A \) and \( B \), then the length of \( \overline{AB} \) is:
\[ \overline{AB} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
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Calculating \( \overline{BC} \): If \( B(x_2, y_2) \) and \( C(x_3, y_3) \) are the coordinates of points \( B \) and \( C \), then the length of \( \overline{BC} \) is:
\[ \overline{BC} = \sqrt{(x_3 - x_2)^2 + (y_3 - y_2)^2} \]
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Using the HL Congruence Theorem: The Hypotenuse-Leg (HL) Congruence Theorem states that if the hypotenuse and one leg of a right triangle are equal to the hypotenuse and one leg of another right triangle, then the triangles are congruent.
Thus, to apply the HL Congruence Theorem to triangles \( ABC \) and \( PQR \), we need:
- The length of the hypotenuse \( \overline{PQ} \) to be equal to \( \overline{AB} \) (the hypotenuse of triangle \( ABC \)).
- The length of one leg (either \( \overline{BC} \) or the other leg of triangle \( PQR \)) to be equal to either \( \overline{BC} \) or the other leg of triangle \( ABC \).
Conclusion: To finalize your answer, you need to provide specific coordinates for points \( A \), \( B \), \( C \), \( P \), \( Q \), and \( R \) to compute \( \overline{AB} \), \( \overline{BC} \), and subsequently \( \overline{PQ} \). Please provide these coordinates to receive specific numerical results.
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