To determine whether triangle \( \Delta ABC \) is equilateral or isosceles, we can calculate the lengths of each side using the distance formula. The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Let's calculate the lengths of sides \( AB \), \( BC \), and \( AC \).
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Calculate AB: \[ AB = \sqrt{(4 - 3)^2 + (4 - 1)^2} = \sqrt{(1)^2 + (3)^2} = \sqrt{1 + 9} = \sqrt{10} \]
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Calculate BC: \[ BC = \sqrt{(5 - 4)^2 + (1 - 4)^2} = \sqrt{(1)^2 + (-3)^2} = \sqrt{1 + 9} = \sqrt{10} \]
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Calculate AC: \[ AC = \sqrt{(5 - 3)^2 + (1 - 1)^2} = \sqrt{(2)^2 + (0)^2} = \sqrt{4} = 2 \]
Now we have the lengths of all three sides:
- \( AB = \sqrt{10} \)
- \( BC = \sqrt{10} \)
- \( AC = 2 \)
Next, we analyze the triangle:
- The lengths \( AB \) and \( BC \) are both \( \sqrt{10} \), and \( AC \) is \( 2 \).
Since \( AB \) and \( BC \) are congruent, the triangle is isosceles. For the triangle to be equilateral, all three sides must be equal, which is not the case here.
Thus, Verna is correct: the triangle is isosceles because \( AB \) is congruent to \( BC \). Therefore, the answer is:
Verna, because segment AB ≅ segment BC.
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