Question

Triangle
\[\triangle A'B'C'\] is the image of
\[\triangle ABC\] under a dilation with a scale factor of
\[2\].
A first quadrant coordinate plane. A triange has vertices A prime at zero, ten, B at four, two, and C at zero, zero.
\[A'\]
\[B'\]
\[C'\]
What is the length of segment

\[\overline{AC}\]?

Answer 1

To find the length of segment \(\overline{AC}\) in triangle \(\triangle ABC\), we first need the coordinates of points \(A\) and \(C\).

From the problem statement, we have:
- \(A\) is at the coordinates \(A(4, 2)\)
- \(C\) is at the coordinates \(C(0, 0)\)

We can calculate the length of segment \(\overline{AC}\) using the distance formula, which is given by:

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Substituting the coordinates of points \(A(4, 2)\) and \(C(0, 0)\) into the distance formula:

\[
d = \sqrt{(0 - 4)^2 + (0 - 2)^2}
\]
\[
d = \sqrt{(-4)^2 + (-2)^2}
\]
\[
d = \sqrt{16 + 4}
\]
\[
d = \sqrt{20}
\]
\[
d = 2\sqrt{5}
\]

Thus, the length of segment \(\overline{AC}\) is \(\boxed{2\sqrt{5}}\).