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Triangle ABC and triangle PQR are plotted in the coordinate plane. Find the lengths of AB¯¯¯¯¯¯¯¯ and BC¯¯¯¯¯¯¯¯, then state the length of PQ¯¯¯¯¯¯¯¯ needed to prove that triangle ABC and triangle PQR are congruent by the HL Congruence Theorem.

AB=$$, BC=$$, PQ=$$

1 answer

To determine whether triangles ABC and PQR are congruent by the HL Congruence Theorem (Hypotenuse-Leg Congruence Theorem), we first need to:

  1. Find the lengths of sides \( AB \) and \( BC \).
  2. Find the length of side \( PQ \).

The HL theorem states that if a triangle has a right angle and you know the lengths of the hypotenuse and one leg, then you can potentially prove that the two triangles are congruent.

Step 1: Finding the Lengths of \( AB \) and \( BC \)

Assuming you have points for triangle \( ABC \):

  • Let \( A(x_1, y_1) \)
  • Let \( B(x_2, y_2) \)
  • Let \( C(x_3, y_3) \)

  1. The length of \( AB \) can be calculated using the distance formula: \[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

  2. The length of \( BC \) is: \[ BC = \sqrt{(x_3 - x_2)^2 + (y_3 - y_2)^2} \]

Step 2: Finding the Length of \( PQ \)

Similarly, for triangle \( PQR \):

  • Let \( P(a_1, b_1) \)
  • Let \( Q(a_2, b_2) \)
  • Let \( R(a_3, b_3) \)
  1. The length of \( PQ \): \[ PQ = \sqrt{(a_2 - a_1)^2 + (b_2 - b_1)^2} \]

Step 3: Conclusion

To state the lengths correctly:

  • If \( AB \) and \( BC \) were found to equal certain values, you must find the corresponding side \( PQ \) that must equal \( AB \) or \( BC \) (whichever leg \( PQ \) corresponds to).

The length of \( PQ \) should be equal to either \( AB \) or \( BC \) for the two triangles (assuming they both have right angles) to be congruent by the HL theorem.

For a complete solution, you would need the specific coordinates for the points \( A, B, C, P, Q, R \). If you provide those coordinates, I can calculate the lengths precisely.