Question

the side lengths of triangle abc are 3, 4, and 5. which set of ordered pairs form a triangle that is congruent to triangle abc?
(-3,1),(-3,5),(0,5)
(1,-2),(1,3),(4,3)
(6,-3),(1,-3),(6,1)
(1,3),(1,4),(5,1)

Explanation:

Step1: Calculate the distance between points in the first option

Use the distance formula \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
For \((-3,1)\) and \((-3,5)\): \(d_1=\sqrt{(-3+3)^2+(5 - 1)^2}=\sqrt{0 + 16}=4\).
For \((-3,5)\) and \((0,5)\): \(d_2=\sqrt{(0 + 3)^2+(5 - 5)^2}=\sqrt{9+0}=3\).
For \((-3,1)\) and \((0,5)\): \(d_3=\sqrt{(0 + 3)^2+(5 - 1)^2}=\sqrt{9 + 16}=5\).

Step2: Check congruence

Since the side - lengths of the triangle formed by \((-3,1),(-3,5),(0,5)\) are \(3\), \(4\), and \(5\), which are the same as the side - lengths of triangle \(ABC\).

Answer:

\((-3,1),(-3,5),(0,5)\)