Question

jessie draws triangle abc on a coordinate grid. the slope of line segment ab is $\frac{3}{4}$. jessie then transforms triangle abc using a single transformation to create triangle abc. she claims the slope of ab will still be $\frac{3}{4}$. for each transformation described, indicate whether it supports or does not support jessies claim. rotation of $180^{circ}$ around the origin reflection across the line $y = 2$ translation up 1.25 units reflection across the $x - axis$

Explanation:

• Rotation of \(180^{\circ}\) around the origin: Rotation is a rigid transformation. Rigid transformations preserve the shape and size of the figure, and also the slope of line segments. If \(AB\) has a slope of \(\frac{3}{4}\), \(A'B'\) (after \(180^{\circ}\) rotation) will have the same slope.
• Reflection across the line \(y = 2\): Reflection is a rigid transformation. A reflection across a horizontal line \(y = k\) (in this case \(k = 2\)) will not change the slope of a non - vertical and non - horizontal line segment. The \(x\) - values of the points on the line segment change in a way that the ratio of the change in \(y\) to the change in \(x\) (the slope) remains the same.
• Translation up \(1.25\) units: Translation is a rigid transformation. A translation (shifting the figure up, down, left, or right) does not change the slope of a line segment. The change in \(y\) and \(x\) values for any two points on the line segment is consistent, so the slope \(\frac{\Delta y}{\Delta x}\) remains \(\frac{3}{4}\).
• Reflection across the \(x\) - axis: Let a point \(A(x_1,y_1)\) and \(B(x_2,y_2)\) on \(AB\) with slope \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{3}{4}\). After reflection across the \(x\) - axis, the points become \(A'(x_1,-y_1)\) and \(B'(x_2,-y_2)\). The slope of \(A'B'\) is \(m'=\frac{-y_2-(-y_1)}{x_2 - x_1}=\frac{-(y_2 - y_1)}{x_2 - x_1}=-\frac{3}{4}

eq\frac{3}{4}\).

Answer:

• Rotation of \(180^{\circ}\) around the origin: Supports Jessie’s Claim
• Reflection across the line \(y = 2\): Supports Jessie’s Claim
• Translation up \(1.25\) units: Supports Jessie’s Claim
• Reflection across the \(x\) - axis: Does Not Support Jessie’s Claim