Question

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

Explanation:

Step1: Analyze the orientation of the triangles

• The orientation of \(\triangle ABC\) and \(\triangle A''B''C''\) is opposite. A \(180^{\circ}\) rotation about the origin changes the orientation of a figure (a reflection over the \(x -\)axis also changes the orientation, but let's check the scale factor next).

Step2: Calculate the scale factor

• Let's assume a vertex. For example, if we consider the length of a side. Suppose in \(\triangle ABC\), if we take \(AC\). Let \(A=(- 10,2)\) and \(C=(0,2)\), so \(AC = 10\) units. In \(\triangle A''B''C''\), if \(A''=(2,-1)\) and \(C''=(-2,-1)\), then \(A''C''=\frac{10}{3}\) units.
• The scale factor \(k\) of a dilation is given by the formula \(k=\frac{\text{length of side in image}}{\text{length of side in pre - image}}\).
• Let's use another approach. If we consider the coordinates. Suppose \(A(-10,2)\), after a \(180^{\circ}\) rotation about the origin \(A\) becomes \((10, - 2)\). But we want to get to \(A''(2,-1)\).
• The scale factor \(k\) for a dilation \((x,y)\to(kx,ky)\). If we assume the pre - image after rotation is \((x,y)\) and the image is \((kx,ky)\). If we take \(x = 6\) (for example, if we consider the horizontal distance from the \(y\) - axis in the original triangle) and \(kx = 2\), then \(k=\frac{1}{3}\).

Answer:

a \(180^{\circ}\) rotation about the origin, then a dilation by a scale factor of \(\frac{1}{3}\)