Quadrilateral

Question

\[A'B'C'D'\] is the image of quadrilateral
\[ABCD\] under a dilation with a scale factor of
\[3\].
A first quadrant coordinate plane. A quadrilateral has vertices A prime at zero, six, B prime at three, zero, C prime at twelve, zero and D prime at zero, nine.
\[A'\]
\[B'\]
\[C'\]
\[D'\]
What is the length of segment

\[\overline{BC}\]?

Answer 1

To find the length of segment \(\overline{BC}\) in the original quadrilateral \(ABCD\), we first need to determine the coordinates of points \(B\) and \(C\) by using the given scale factor and the coordinates of the points in the dilated quadrilateral \(A'B'C'D'\).

Given the dilation scale factor is \(3\), the coordinates of each point in \(ABCD\) can be found by taking the corresponding coordinates of points in \(A'B'C'D'\) and dividing each coordinate by \(3\).

Let's calculate the points:

• \(B'\) has coordinates \((3, 0)\).

  • The coordinates of point \(B\) in quadrilateral \(ABCD\) will be: \[ B = \left(\frac{3}{3}, \frac{0}{3}\right) = (1, 0) \]

• \(C'\) has coordinates \((12, 0)\).

  • The coordinates of point \(C\) in quadrilateral \(ABCD\) will be: \[ C = \left(\frac{12}{3}, \frac{0}{3}\right) = (4, 0) \]

Now, we have the coordinates:

• Point \(B\) is at \((1, 0)\)
• Point \(C\) is at \((4, 0)\)

To find the length of segment \(\overline{BC}\), we can use the distance formula. However, since both points are on the same horizontal line (y = 0), the length of the segment can be simply computed as the absolute difference in their x-coordinates:

\[ \text{Length of } \overline{BC} = |x_C - x_B| = |4 - 1| = 3 \]

Thus, the length of segment \(\overline{BC}\) is \(\boxed{3}\).