To determine whether [tex]\( \triangle ABC \)[/tex] is an isosceles triangle, a right triangle, or both, we need to analyze the coordinates of the vertices of the triangle. The coordinates are:
[tex]\[
A = (2, 0), \quad B = (4, 4), \quad C = (6, 3)
\][/tex]
Step 1: Calculate the lengths of the sides using the distance formula.
The distance formula for two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Calculating [tex]\( AB \)[/tex]:
[tex]\[
AB = \sqrt{(4-2)^2 + (4-0)^2} = \sqrt{2^2 + 4^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5}
\][/tex]
Calculating [tex]\( BC \)[/tex]:
[tex]\[
BC = \sqrt{(6-4)^2 + (3-4)^2} = \sqrt{2^2 + (-1)^2} = \sqrt{4 + 1} = \sqrt{5}
\][/tex]
Calculating [tex]\( AC \)[/tex]:
[tex]\[
AC = \sqrt{(6-2)^2 + (3-0)^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5
\][/tex]
Step 2: Determine if the triangle is isosceles by checking if any two sides are equal.
[tex]\[
AB = 2\sqrt{5}, \quad BC = \sqrt{5}, \quad AC = 5
\][/tex]
No two sides are equal. Therefore, [tex]\( \triangle ABC \)[/tex] is not an isosceles triangle.
Step 3: Determine if the triangle is a right triangle by checking the dot product of vectors.
First, define the vectors:
[tex]\[
\overrightarrow{AB} = (B_x - A_x, B_y - A_y) = (4 - 2, 4 - 0) = (2, 4)
\][/tex]
[tex]\[
\overrightarrow{BC} = (C_x - B_x, C_y - B_y) = (6 - 4, 3 - 4) = (2, -1)
\][/tex]
[tex]\[
\overrightarrow{AC} = (C_x - A_x, C_y - A_y) = (6 - 2, 3 - 0) = (4, 3)
\][/tex]
Calculate the dot products:
[tex]\[
\overrightarrow{AB} \cdot \overrightarrow{BC} = (2, 4) \cdot (2, -1) = (2)(2) + (4)(-1) = 4 - 4 = 0
\][/tex]
Since the dot product [tex]\(\overrightarrow{AB} \cdot \overrightarrow{BC} = 0\)[/tex], the vectors [tex]\(\overrightarrow{AB}\)[/tex] and [tex]\(\overrightarrow{BC}\)[/tex] are perpendicular. Therefore, [tex]\( \triangle ABC \)[/tex] is a right triangle with [tex]\( \overline{AB} \)[/tex] perpendicular to [tex]\( \overline{BC} \)[/tex].
Conclusion:
Since [tex]\( \triangle ABC \)[/tex] is a right triangle and not an isosceles triangle, Verna is correct.
Therefore, the answer is:
[tex]\[
\boxed{4} \quad \text{Verna, because} \ \overline{AB} \perp \overline{BC}
\][/tex]
