Triangle ABC

Question

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=

Answer 1

To find the lengths \( AB \) and \( BC \) for triangle \( ABC \) and the length \( PQ \) for triangle \( PQR \) needed to prove that the two triangles are congruent by the HL (Hypotenuse-Leg) Congruence Theorem, we would typically follow these steps:

1. Calculate lengths using the distance formula: The distance formula between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) in the coordinate plane is given by:

   \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

2. Identify the coordinates of points: Without specific coordinates for points \( A \), \( B \), \( C \), \( P \), \( Q \), and \( R \), I cannot calculate the exact lengths. Please provide the coordinates for the points \( A \), \( B \), \( C \), \( P \), \( Q \), and \( R \) in order to perform these calculations.

3. Verify the right triangle condition (necessary for HL congruence): Ensure that at least one triangle is a right triangle (check that it satisfies the Pythagorean theorem).

4. Length of \( PQ \): After calculating \( AB \) and \( BC \), the length \( PQ \) needed is simply the length of the hypotenuse of triangle \( PQR \).

Please provide the specific coordinates to proceed with the calculations.