Questions: Triangle PQR was transformed to create the congruent triangle STU. Which describes how triangle PQR could have been transformed? reflected across the y-axis and dilated by a scale factor of 1/2 dilated by a scale factor of 2 and translated 3 units to the right translated 2 units down and dilated by a scale factor of 3/2 rotated 90° about the origin and reflected across the x-axis

Triangle PQR was transformed to create the congruent triangle STU. Which describes how triangle PQR could have been transformed?
reflected across the y-axis and dilated by a scale factor of 1/2
dilated by a scale factor of 2 and translated 3 units to the right
translated 2 units down and dilated by a scale factor of 3/2
rotated 90° about the origin and reflected across the x-axis

Transcript text: Triangle $P Q R$ was transformed to create the congruent triangle $S T U$. Which describes how triangle $P Q R$ could have been transformed? reflected across the $y$-axis and dilated by a scale factor of $\frac{1}{2}$ dilated by a scale factor of 2 and translated 3 units to the right translated 2 units down and dilated by a scale factor of $\frac{3}{2}$ rotated $90^{\circ}$ about the origin and reflected across the $x$-axis

Solution

Solution Steps

To determine which transformation describes how triangle $PQR$ could have been transformed to create the congruent triangle $STU$, we need to consider the properties of congruent triangles. Congruent triangles have the same shape and size, but their positions and orientations may differ. Therefore, any transformation that changes the size of the triangle (dilation) cannot be correct. We need to identify a transformation that preserves the size and shape of the triangle.

Solution Approach

Reflection across the y-axis and dilation by a scale factor of $\frac{1}{2}$: This transformation changes the size of the triangle, so it cannot be correct.
Dilation by a scale factor of 2 and translation 3 units to the right: This transformation changes the size of the triangle, so it cannot be correct.
Translation 2 units down and dilation by a scale factor of $\frac{3}{2}$: This transformation changes the size of the triangle, so it cannot be correct.
Rotation $90^\circ$ about the origin and reflection across the x-axis: This transformation preserves the size and shape of the triangle, so it is a possible correct answer.

Step 1: Define the Original Points

Given the points of triangle $PQR$:

$P = (1, 2)$
$Q = (3, 4)$
$R = (5, 6)$

Step 2: Apply $90^\circ$ Rotation About the Origin

To rotate a point $(x, y)$ by $90^\circ$ about the origin, the new coordinates are $(-y, x)$:

$P_{\text{rotated}} = (-2, 1)$
$Q_{\text{rotated}} = (-4, 3)$
$R_{\text{rotated}} = (-6, 5)$

Step 3: Reflect Across the x-axis

To reflect a point $(x, y)$ across the x-axis, the new coordinates are $(x, -y)$:

$P_{\text{transformed}} = (-2, -1)$
$Q_{\text{transformed}} = (-4, -3)$
$R_{\text{transformed}} = (-6, -5)$

Final Answer

The transformation that describes how triangle $PQR$ could have been transformed to create the congruent triangle $STU$ is: \[ \boxed{\text{Rotated } 90^\circ \text{ about the origin and reflected across the } x\text{-axis}} \]

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