Question

the graph shows triangles abc and stu.
is abc similar to stu? justify your answer.
yes, because a dilation by a scale factor of \\( \frac { 1 } { 2 } \\) centered at point b followed by a rotation \\( 180 ^ { \circ } \\) around the origin maps abc onto stu.
yes, because a dilation by a scale factor of \\( \frac { 1 } { 2 } \\) centered at point a followed by a rotation \\( 180 ^ { \circ } \\) around the origin maps abc onto stu.
no, because \\( \angle b \\) and \\( \angle t \\) do not have the same measure.

Explanation:

Step1: Calculate the side lengths

Use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
For \(\triangle ABC\):

• \(AB=\sqrt{(-3+8)^2+(10 - 6)^2}=\sqrt{25 + 16}=\sqrt{41}\)
• \(BC=\sqrt{(2+3)^2+(-2 - 10)^2}=\sqrt{25+144}=\sqrt{169} = 13\)
• \(AC=\sqrt{(2+8)^2+(-2 - 6)^2}=\sqrt{100 + 64}=\sqrt{164}\)

For \(\triangle STU\):

• \(ST=\sqrt{(5 - 2)^2+(-9+10)^2}=\sqrt{9 + 1}=\sqrt{10}\) (This is wrong approach, we can also use the property of dilation. If we assume a dilation centered at a wrong point, the side - length ratio won't match. But if we check the angles.

Another way: We know that if two triangles are similar, their corresponding angles are equal.
If we consider the transformation. A dilation centered at \(A\) with scale factor \(\frac{1}{2}\) will not map \(ABC\) to \(STU\) correctly.
If we consider the rotation and dilation.
Let's check the angle - angle similarity.
We can also use the fact that if a dilation is centered at \(B\) with scale factor \(\frac{1}{2}\), the position won't be correct.
But if we check the slope of the sides to find angles.
The slope of \(AB\) in \(\triangle ABC\): \(m_{AB}=\frac{10 - 6}{-3+8}=\frac{4}{5}\)
The slope of \(ST\) in \(\triangle STU\): \(m_{ST}=\frac{-9 + 10}{5 - 2}=\frac{1}{3}\) (This is also a wrong way. The correct way is to use the property of similarity transformation.
Since there is no dilation - rotation combination (centered at \(A\) or \(B\)) that can map \(ABC\) to \(STU\) such that the side - length ratios are equal and angles are equal. And we can observe from the graph (by counting the grid units for side - lengths in a non - formal way, assume \(AB\) spans \(5\) units in \(x\) - direction and \(4\) units in \(y\) - direction (approximate from grid), \(ST\) spans \(3\) units in \(x\) - direction and \(1\) unit in \(y\) - direction. The ratios are not equal. Also, if we check the angle \(\angle B\) and \(\angle T\) (by the direction of the sides forming the angles), they are not equal.

Answer:

No, because \(\angle B\) and \(\angle T\) do not have the same measure.