To find the lengths of sides \( AB \) and \( BC \) in triangle \( ABC \) and to determine the length of side \( PQ \) in triangle \( PQR \) for proving congruence by the HL (Hypotenuse-Leg) Congruence Theorem, we need some specific coordinates for points A, B, C, P, Q, and R.
The HL Congruence Theorem states that if the hypotenuse and one leg of one right triangle are equal to the hypotenuse and one leg of another right triangle, then the two triangles are congruent.
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Finding the lengths of \( AB \) and \( BC \):
- Suppose point \( A \) has coordinates \( (x_1, y_1) \), point \( B \) has coordinates \( (x_2, y_2) \), and point \( C \) has coordinates \( (x_3, y_3) \).
- The length of \( AB \) can be calculated using the distance formula: \[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
- The length of \( BC \) can be calculated similarly: \[ BC = \sqrt{(x_3 - x_2)^2 + (y_3 - y_2)^2} \]
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Finding the length of \( PQ \):
- Suppose point \( P \) has coordinates \( (x_4, y_4) \) and point \( Q \) has coordinates \( (x_5, y_5) \).
- The length of \( PQ \) is calculated using the same distance formula: \[ PQ = \sqrt{(x_5 - x_4)^2 + (y_5 - y_4)^2} \]
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Condition for Congruence via HL Theorem:
- For triangles \( ABC \) and \( PQR \) to be congruent by the HL Theorem, the following must hold:
- The lengths of the hypotenuses \( AB \) of triangle \( ABC \) and \( PQ \) of triangle \( PQR \) must be equal.
- One leg (e.g., \( BC \) of triangle \( ABC \)) must equal the corresponding leg of triangle \( PQR \) (which could be \( QR \), depending on its coordinates).
- For triangles \( ABC \) and \( PQR \) to be congruent by the HL Theorem, the following must hold:
Conclusion:
- To derive the lengths, exact coordinates for the triangles are necessary. Please provide the coordinates for points A, B, C for triangle ABC, and points P, Q, R for triangle PQR.
- The lengths of \( AB, BC, \) and \( PQ \) can then be computed, allowing you to state the necessary length for \( PQ \) needed to claim congruence.