To find the lengths of the sides of triangles ABC and PQR, we can use the distance formula, which is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Step 1: Calculate length of AB
Points A and B have coordinates A(1, 5) and B(1, 2):
\[ AB = \sqrt{(1 - 1)^2 + (2 - 5)^2} = \sqrt{0 + (-3)^2} = \sqrt{9} = 3 \]
Step 2: Calculate length of BC
Points B and C have coordinates B(1, 2) and C(5, 2):
\[ BC = \sqrt{(5 - 1)^2 + (2 - 2)^2} = \sqrt{(4)^2 + 0} = \sqrt{16} = 4 \]
Step 3: Calculate length of PQ
Points P and Q have coordinates P(-2, -3) and Q(-3, -3):
\[ PQ = \sqrt{(-3 - (-2))^2 + (-3 - (-3))^2} = \sqrt{(-3 + 2)^2 + 0} = \sqrt{(-1)^2} = 1 \]
Final Results:
- \( AB = 3 \)
- \( BC = 4 \)
- \( PQ = 1 \)
Conclusion
To prove that triangles ABC and PQR are congruent by the HL Congruence Theorem, we need to check that the lengths of the hypotenuses are equal. Given that we have found:
- Length of AB = 3
- Length of BC = 4
For triangle PQR to be congruent to triangle ABC via the HL theorem, \( PQ \) must equal the hypotenuse of triangle ABC. Since triangle ABC is a right triangle (with right angle at B), we can find the length of AC using the Pythagorean theorem:
\[ AC = \sqrt{AB^2 + BC^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
Thus, \( PQ \) must equal 5 for the triangles to be congruent.
So, the answers are:
- \( AB = 3 \)
- \( BC = 4 \)
- \( PQ = 5 \) (needed length)