Question

ABC and PQR are plotted in the coordinate plane. Find the lengths of AB and BC, then state the length of PQ needed to prove that ABC and PQR are congrunt by the HL congruence theorem.

Assume that both the x-axis and y-axis are drawn with intervals of 1. Count the units to find the legnths of the sides of ABC.

AB = ____, BC = ____, PQ = ____

Answer 1

To solve for the lengths of sides AB, BC, and PQ, we need the coordinates of the points A, B, C, P, Q, and R for triangles ABC and PQR. Since these coordinates are not provided in your question, I cannot calculate the lengths directly.

However, assuming you have the coordinates, here's how you can calculate the lengths of the sides using the distance formula:

1. **Calculate the length of AB**:
If \( A(x_1, y_1) \) and \( B(x_2, y_2) \), then:
\[
AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

2. **Calculate the length of BC**:
If \( B(x_2, y_2) \) and \( C(x_3, y_3) \), then:
\[
BC = \sqrt{(x_3 - x_2)^2 + (y_3 - y_2)^2}
\]

3. **For the length of PQ needed for congruence by the HL theorem**:
The Hypotenuse-Leg (HL) theorem states that in right triangles, if the length of the hypotenuse and one leg of one triangle are equal to the lengths of the hypotenuse and the corresponding leg of another triangle, then the triangles are congruent. Thus, to prove triangles ABC and PQR congruent by the HL theorem, you would need to find the length of the hypotenuse (which could be AC or PR, depending on the configuration of points) in triangle ABC and equate it with the corresponding hypotenuse in triangle PQR.

Please provide the coordinates or more specific information regarding the triangles, and I'd be happy to assist you in calculating those lengths!