Question

the sides \\( \overline { l m } \\), \\( \overline { m n } \\), and \\( \overline { l n } \\) of \\( \triangle l m n \\) have lengths of 3, 4, and 5, respectively. which triangle is congruent to \\( \triangle l m n \\)?

Explanation:

Step1: Calculate side lengths using distance formula

The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
For the third triangle:
Let's assume two vertices \((-1,3)\) and \((-1, - 1)\), then \(d_1=\sqrt{(-1+1)^2+( - 1 - 3)^2}=\sqrt{0 + 16}=4\).
Let's assume two vertices \((-1,-1)\) and \((2,-1)\), then \(d_2=\sqrt{(2 + 1)^2+(-1 + 1)^2}=\sqrt{9+0}=3\).
Let's assume two vertices \((-1,3)\) and \((2,-1)\), then \(d_3=\sqrt{(2 + 1)^2+(-1 - 3)^2}=\sqrt{9 + 16}=\sqrt{25}=5\).

Answer:

The third triangle (the one with side - lengths calculated as 3,4,5 using the distance formula) is congruent to \(\triangle LMN\)