Question

angle xyz is shown in the xy - plane. vertex x has coordinates. the length of xy is 2 units, and the length of xz is 6 units. students work for finding the slope of the perpendicular bisector of yz is shown. students work: the slope of yz is 2/6, or 1/3. the opposite of the reciprocal of 1/3 is - 3. so, the perpendicular bisector of yz has a slope of - 3. describe the students mistake. find the equation of the line that represents the perpendicular bisector of yz. show your work or explain how you found the equation. enter your answer and your work or explanation in the space provided. you may also use the drawing tool to help explain or support your answer.

Explanation:

Step1: Identify the student's mistake

The student incorrectly calculated the slope of $\overline{YZ}$. Since $Y$ is 2 units above $X$ and $Z$ is 6 units to the right of $X$, the slope of $\overline{YZ}$ is $\frac{2 - 0}{0 - 6}=-\frac{1}{3}$ (assuming $Y$ has the same $x -$coordinate as $X$ and $Z$ has the same $y -$coordinate as $X$ for the right - triangle formed). The opposite of the reciprocal of $-\frac{1}{3}$ is 3, not - 3.

Step2: Find the mid - point of $\overline{YZ}$

The coordinates of $Y$ are $(2,5)$ (2 units above $X(2,3)$) and the coordinates of $Z$ are $(8,3)$ (6 units to the right of $X(2,3)$). The mid - point formula is $(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})$. For $Y(2,5)$ and $Z(8,3)$, the mid - point $M$ is $(\frac{2 + 8}{2},\frac{5+3}{2})=(5,4)$.

Step3: Find the equation of the perpendicular bisector

The slope of the perpendicular bisector of $\overline{YZ}$ is 3 (from Step 1). Using the point - slope form of a line $y - y_1=m(x - x_1)$, where $m = 3$ and $(x_1,y_1)=(5,4)$. Substitute into the formula: $y - 4=3(x - 5)$. Expand to get $y-4 = 3x-15$, and then $y=3x - 11$.

Answer:

The student's mistake is incorrectly calculating the slope of $\overline{YZ}$. The correct slope of the perpendicular bisector of $\overline{YZ}$ is 3. The equation of the perpendicular bisector of $\overline{YZ}$ is $y = 3x-11$.