Question

triangle xyz with vertices x(5, 7), y(8, 3), and z(2, 3) is reflected over the y - axis and translated up 4 units to form triangle xyz. what is the length of segment
a. 2 units
b. 4 units
c. 6 units
d. 10 units

Explanation:

Step1: Recall transformation properties

Reflections and translations are rigid - motions. Rigid - motions preserve the lengths of segments. So, the length of segment in the original triangle \(XYZ\) is the same as the length of the corresponding segment in the transformed triangle \(X'Y'Z'\).

Step2: Use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\)

Let's assume we want to find the length of a side, say between two points \(X(5,7)\) and \(Y(6,3)\).
\(x_1 = 5,y_1 = 7,x_2 = 6,y_2 = 3\)
\[

$$\begin{align*} d&=\sqrt{(6 - 5)^2+(3 - 7)^2}\\ &=\sqrt{1^2+( - 4)^2}\\ &=\sqrt{1 + 16}\\ &=\sqrt{17}\approx 4.12 \end{align*}$$

\]
Let's assume we calculate the distance between \(Y(6,3)\) and \(Z(2,3)\)
\(x_1 = 6,y_1 = 3,x_2 = 2,y_2 = 3\)
\[

$$\begin{align*} d&=\sqrt{(2 - 6)^2+(3 - 3)^2}\\ &=\sqrt{( - 4)^2+0^2}\\ &=\sqrt{16}\\ & = 4 \end{align*}$$

\]

Answer:

B. 4 units