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Right triangle upper A upper B upper C is plotted in the first quadrant of a coordinate plane with unlabeled axes. Right triangle upper P upper Q upper R is plotted in the third quadrant. Squares are drawn in the interior angles by upper Q and upper B.

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.

(2 points)
AB=
, BC=
, PQ=

1 answer

To solve this problem, let's define the right triangles and their sides based on their properties and positions in the coordinate plane.

Given:

  • Triangle ABC is a right triangle in the first quadrant.
  • Triangle PQR is a right triangle in the third quadrant.
  • We need to find the lengths of \( \overline{AB} \), \( \overline{BC} \), and the length of \( \overline{PQ} \) to show congruence by the HL (Hypotenuse-Leg) Congruence Theorem.

Step 1: Define the triangles

  1. Triangle ABC:

    • Let \( A = (0, 0) \)
    • Let \( B = (a, 0) \) (base of the triangle on the x-axis)
    • Let \( C = (a, b) \) (height on the y-axis)

    Using the coordinates, we can find the lengths:

    • \( AB = a \) (length on the x-axis)
    • \( BC = b \) (length on the y-axis)

  2. Triangle PQR:

    • Triangle PQR is in the third quadrant. Let:
      • \( P = (x, y) \)
      • \( Q = (x + a, y) \) (corresponding to B)
      • \( R = (x + a, y - b) \) (corresponding to C)

    Using the coordinates for PQR:

    • \( PQ = a \) (base)
    • \( PR = b \) (height)

Step 2: Identify required lengths

  • From the conditions of the problem, we find:
    • \( \overline{AB} = a \)
    • \( \overline{BC} = b \)
    • \( \overline{PQ} = a \) (as there is a corresponding segment)

Conclusion

By the HL Congruence Theorem:

  • If two triangles have a hypotenuse that is equal in length and one leg that is also equal in length, then the triangles are congruent.
  • Here, both triangles ACB and PQR have corresponding legs equal, provided \( a \) and \( b \) are the same in both triangles.

Thus:

  • \( AB = a \) (length of AB)
  • \( BC = b \) (length of BC)
  • \( PQ = a \)

If specific numeric values are needed for \( AB \) and \( BC \), please provide more information about the coordinates of points or any additional details, as the values of \( a \) and \( b \) are currently left in a general form.