Question

on a math test ali needs to describe the motions which map \\( \triangle efg \\) to \\( \triangle mno \\). the coordinates of \\( \triangle efg \\) are \\( e(5,5) \\), \\( f(7,5) \\), and \\( g(7,10) \\). the coordinates of \\( \triangle mno \\) are \\( m(0,0) \\), \\( n(-2,0) \\), and \\( o(-2,5) \\). he says there is a translation 5 units to the left and 5 units down, followed by a reflection across the \\( x \\)-axis. describe the sequence of rigid motions. what is alis likely error?
which of the following is the description of the rigid motions which maps \\( \triangle efg \\) to \\( \triangle mno \\)?
a. a translation 5 units to the left and 5 units down, followed by a reflection across the \\( y \\)-axis
b. a translation 4 units to the left and 4 units down, followed by a reflection across the \\( y \\)-axis
c. a translation 5 units to the left and 5 units down, followed by a reflection across the \\( x \\)-axis
what is alis likely error?
a. he did not move the correct units to the left.
b. he reflected the triangle across the wrong axis.
c. he did not move the correct units down.

Explanation:

Step1: Analyze the translation

For point \(E(5,5)\) to get to a position before reflection:
If we consider the correct mapping, after translation \(5\) units left (\(x - 5\)) and \(5\) units down (\(y-5\)), \(E(5,5)\) becomes \((5 - 5,5 - 5)=(0,0)\). But if we then reflect \((0,0)\) across the \(x -\)axis, it remains \((0,0)\). However, if we first do the translation and then reflect across the \(y-\)axis:
Let's assume we have a general point \((x,y)\). After translation \(T(x,y)=(x - 5,y - 5)\), and then reflection across the \(y-\)axis \(R_y(x,y)=(-x,y)\).
For \(E(5,5)\): \(T(5,5)=(0,0)\), \(R_y(0,0)=(0,0)\); for \(F(7,5)\): \(T(7,5)=(2,0)\), \(R_y(2,0)=(- 2,0)\); for \(G(7,10)\): \(T(7,10)=(2,5)\), \(R_y(2,5)=(-2,5)\) which matches \(\triangle MNO\) with \(M(0,0)\), \(N(-2,0)\), \(O(-2,5)\)

Step2: Check Ali's reflection axis

Ali said “a translation \(5\) units to the left and \(5\) units down, followed by a reflection across the \(x -\)axis”.
The formula for reflection across the \(x -\)axis is \(R_x(x,y)=(x,-y)\). If we take the point after translation \(T(5,5)=(0,0)\), \(R_x(0,0)=(0,0)\); \(T(7,5)=(2,0)\), \(R_x(2,0)=(2,0)\) (does not match \(N(-2,0)\)); \(T(7,10)=(2,5)\), \(R_x(2,5)=(2, - 5)\) (does not match \(O(-2,5)\))

Answer:

A. A translation 5 units to the left and 5 units down, followed by a reflection across the y - axis
B. He reflected the triangle across the wrong axis.