Question

which transformations could be performed to show that △abc is similar to △abc*?
a reflection over the x - axis, then a dilation by a scale factor of 3
a reflection over the x - axis, then a dilation by a scale factor of \\( \frac { 1 } { 3 } \\)
a 180° rotation about the origin, then a dilation by a scale factor of 3
a 180° rotation about the origin, then a dilation by a scale factor of \\( \frac { 1 } { 3 } \\)

Explanation:

Step 1: Find the coordinates of the vertices

Coordinates of \(\triangle ABC\): \(A(-10,3)\), \(B(-10,6)\), \(C(0,3)\)
Coordinates of \(\triangle A''B''C''\): \(A''(2, - 1)\), \(B''(2,-2)\), \(C''(-2,-1)\)

Step 2: Analyze the transformation

If we first consider a reflection over the \(x -\)axis. The rule for a reflection over the \(x -\)axis is \((x,y)\to(x, - y)\).
For point \(A(-10,3)\) after reflection over the \(x -\)axis: \((-10,-3)\)
For point \(B(-10,6)\) after reflection over the \(x -\)axis: \((-10,-6)\)
For point \(C(0,3)\) after reflection over the \(x -\)axis: \((0,-3)\)

Now, if we use the dilation formula \((x,y)\to(kx,ky)\). Let's check the scale factor \(k\).
Take a pair of corresponding points. Let's take the reflected point (say \((- 10,-3)\)) and the final point \(A''(2,-1)\)
We know that \(kx = 2\) and \(ky=-1\) if the original \(x=-10\) (after reflection \(x=-10\), \(y = - 3\))
If we assume the transformation is reflection over \(x -\)axis followed by dilation.
Let's use the ratio of the \(x\) - coordinates (or \(y\) - coordinates).
\(k=\frac{2}{-10}\times(-1)=\frac{1}{3}\) (using the \(x\) - coordinate of \(A\) after reflection \((x=-10)\) and \(A''(x = 2)\))

Answer:

a reflection over the \(x\) - axis, then a dilation by a scale factor of \(\frac{1}{3}\)