{ \triangle XYZ$}$ Was Reflected Over A Vertical Line, Then Dilated By A Scale Factor Of { \frac{1}{2}$}$, Resulting In { \triangle X {\prime}Y {\prime}Z^{\prime}$}$. Which Must Be True Of The Two Triangles? Select Three

Reflection and Dilation of Triangles: Understanding the Relationship Between $\triangle XYZ$ and $\triangle X^{\prime}Y^{\prime}Z^{\prime}$

In geometry, transformations play a crucial role in understanding the properties and relationships between shapes. Reflection and dilation are two fundamental transformations that can be applied to triangles, resulting in new triangles with distinct properties. In this article, we will explore the effects of reflecting $\triangle XYZ$ over a vertical line and then dilating it by a scale factor of $\frac{1}{2}$, resulting in $\triangle X^{\prime}Y^{\prime}Z^{\prime}$. We will examine the properties of the two triangles and determine which statements must be true of the two triangles.

When $\triangle XYZ$ is reflected over a vertical line, the resulting triangle, $\triangle X^{\prime}Y^{\prime}Z^{\prime}$, is a mirror image of the original triangle. This means that the x-coordinates of the vertices of $\triangle X^{\prime}Y^{\prime}Z^{\prime}$ are the negative of the x-coordinates of the vertices of $\triangle XYZ$. The reflection preserves the size and shape of the triangle, but it changes the orientation of the triangle.

After reflecting $\triangle XYZ$ over a vertical line, we dilate $\triangle X^{\prime}Y^{\prime}Z^{\prime}$ by a scale factor of $\frac{1}{2}$. This means that the distance between each vertex of $\triangle X^{\prime}Y^{\prime}Z^{\prime}$ and the origin is reduced by a factor of $\frac{1}{2}$. As a result, the size of the triangle is reduced, but its shape remains the same.

Now that we have reflected and dilated $\triangle XYZ$, we can examine the properties of the two triangles. The following statements must be true of the two triangles:

1. The two triangles are similar

Since $\triangle X^{\prime}Y^{\prime}Z^{\prime}$ is a dilation of $\triangle XYZ$ by a scale factor of $\frac{1}{2}$, the two triangles are similar. This means that the corresponding angles of the two triangles are equal, and the corresponding sides are proportional.

2. The two triangles have the same shape

As mentioned earlier, the reflection and dilation transformations preserve the shape of the triangle. Therefore, $\triangle XYZ$ and $\triangle X^{\prime}Y^{\prime}Z^{\prime}$ have the same shape.

3. The two triangles have different sizes

The dilation transformation reduces the size of $\triangle X^{\prime}Y^{\prime}Z^{\prime}$ by a factor of $\frac{1}{2}$. Therefore, the two triangles have different sizes.

4. The two triangles have the same orientation

Since the reflection transformation preserves the orientation of the triangle, $\triangle XYZ$ and $\triangle X^{\prime}Y^{\prime}Z^{\prime}$ have the same orientation.

5. The two triangles have different positions

The reflection and dilation transformations change the position of $\triangle X^{\prime}Y^{\prime}Z^{\prime}$ relative to $\triangle XYZ$. Therefore, the two triangles have different positions.

In conclusion, when $\triangle XYZ$ is reflected over a vertical line and then dilated by a scale factor of $\frac{1}{2}$, resulting in $\triangle X^{\prime}Y^{\prime}Z^{\prime}$, the following statements must be true of the two triangles:

• The two triangles are similar.
• The two triangles have the same shape.
• The two triangles have different sizes.
• The two triangles have the same orientation.
• The two triangles have different positions.

In this article, we will address some of the most common questions related to the reflection and dilation of triangles.

Q: What is the difference between reflection and dilation?

A: Reflection is a transformation that flips a shape over a line, while dilation is a transformation that changes the size of a shape.

Q: How does reflection affect the size and shape of a triangle?

A: Reflection preserves the size and shape of a triangle, but it changes the orientation of the triangle.

Q: How does dilation affect the size and shape of a triangle?

A: Dilation changes the size of a triangle, but it preserves the shape of the triangle.

Q: What is the scale factor in dilation?

A: The scale factor in dilation is the ratio of the distance between the vertices of the dilated triangle and the distance between the vertices of the original triangle.

Q: What is the effect of a scale factor of 1 on a triangle?

A: A scale factor of 1 has no effect on a triangle, as it means that the distance between the vertices of the dilated triangle is the same as the distance between the vertices of the original triangle.

Q: What is the effect of a scale factor of 0 on a triangle?

A: A scale factor of 0 means that the dilated triangle is a point, as the distance between the vertices of the dilated triangle is 0.

Q: Can a triangle be dilated by a scale factor of -1?

A: Yes, a triangle can be dilated by a scale factor of -1, but it would be equivalent to reflecting the triangle over a line and then dilating it by a scale factor of 1.

Q: What is the relationship between the two triangles after dilation?

A: The two triangles are similar, as the corresponding angles are equal and the corresponding sides are proportional.

Q: Can a triangle be dilated by a scale factor of 2?

A: Yes, a triangle can be dilated by a scale factor of 2, which means that the distance between the vertices of the dilated triangle is twice the distance between the vertices of the original triangle.

Q: What is the effect of dilating a triangle by a scale factor of 2 and then reflecting it over a line?

A: The effect of dilating a triangle by a scale factor of 2 and then reflecting it over a line is the same as dilating the triangle by a scale factor of 2.

In conclusion, the reflection and dilation of triangles are fundamental concepts in geometry that can be used to understand the properties and relationships between shapes. By understanding the effects of reflection and dilation, we can better appreciate the beauty and complexity of geometric transformations.

• Reflection: A transformation that flips a shape over a line.
• Dilation: A transformation that changes the size of a shape.
• Scale factor: The ratio of the distance between the vertices of the dilated triangle and the distance between the vertices of the original triangle.
• Similar triangles: Triangles that have the same shape and size.
• Proportional sides: Sides of similar triangles that have the same ratio.

• [1] Geometry: A Comprehensive Introduction. Michael Spivak.
• [2] Geometry: A Modern Approach. David Kay.
• [3] Mathematics: A Human Approach. Harold R. Jacobs.

• [1] Khan Academy: Geometry
• [2] Math Open Reference: Geometry
• [3] Wolfram MathWorld: Geometry