7-1 Additional Practice Dilations Answer Key

7-1 Additional Practice: Dilations Answer Key & Comprehensive Guide

This comprehensive guide provides answers and explanations for a hypothetical "7-1 Additional Practice" worksheet focusing on dilations in geometry. Since I don't have access to a specific worksheet titled "7-1 Additional Practice: Dilations," I will create a representative set of problems covering various aspects of dilations. This guide will serve as a valuable resource for students to check their understanding and solidify their knowledge of this important geometric concept.

Understanding Dilations

Before diving into the answer key, let's review the fundamental concept of dilations. A dilation is a transformation that changes the size of a figure, but not its shape. It's defined by a center of dilation and a scale factor.

• Center of Dilation: This is a fixed point around which the figure is enlarged or reduced.
• Scale Factor: This is a number that determines the ratio of the new size to the original size. A scale factor greater than 1 results in an enlargement, while a scale factor between 0 and 1 results in a reduction. A scale factor of 1 results in no change.

The process involves multiplying the coordinates of each point in the original figure by the scale factor. The center of dilation remains unchanged.

Key Properties of Dilations:

• Shape Preservation: The shape of the figure remains the same after dilation.
• Size Change: The size of the figure changes proportionally according to the scale factor.
• Parallel Lines: Parallel lines in the original figure remain parallel after dilation.
• Collinearity: Points that are collinear in the original figure remain collinear after dilation.
• Ratio Preservation: The ratios of corresponding side lengths remain the same after dilation.

Hypothetical 7-1 Additional Practice Problems & Answer Key

Let's consider a set of problems that would typically be found in a "7-1 Additional Practice: Dilations" worksheet.

Problem 1:

A triangle with vertices A(2, 4), B(6, 2), and C(4, 0) is dilated using the origin (0, 0) as the center of dilation and a scale factor of 2. Find the coordinates of the vertices of the dilated triangle A'B'C'.

Answer 1:

To find the coordinates of the dilated triangle, multiply the coordinates of each vertex by the scale factor:

• A'(2 * 2, 4 * 2) = A'(4, 8)
• B'(6 * 2, 2 * 2) = B'(12, 4)
• C'(4 * 2, 0 * 2) = C'(8, 0)

Therefore, the vertices of the dilated triangle are A'(4, 8), B'(12, 4), and C'(8, 0).

Problem 2:

A square with vertices D(1, 1), E(3, 1), F(3, 3), and G(1, 3) is dilated with a center of dilation at (0, 0) and a scale factor of 0.5. What are the coordinates of the vertices of the dilated square?

Answer 2:

Multiply the coordinates of each vertex by the scale factor of 0.5:

• D'(1 * 0.5, 1 * 0.5) = D'(0.5, 0.5)
• E'(3 * 0.5, 1 * 0.5) = E'(1.5, 0.5)
• F'(3 * 0.5, 3 * 0.5) = F'(1.5, 1.5)
• G'(1 * 0.5, 3 * 0.5) = G'(0.5, 1.5)

The vertices of the dilated square are D'(0.5, 0.5), E'(1.5, 0.5), F'(1.5, 1.5), and G'(0.5, 1.5).

Problem 3:

A line segment with endpoints P( -2, 6) and Q(4, -2) is dilated using the point (1, 2) as the center of dilation and a scale factor of 3. Find the coordinates of the dilated endpoints P' and Q'.

Answer 3: This problem requires a slightly different approach because the center of dilation is not the origin. We need to find the displacement of each point from the center of dilation, multiply the displacement by the scale factor, and then add the center back to the result.

• For Point P:

  • Displacement from center: (-2 -1, 6 -2) = (-3, 4)
  • Scaled displacement: (-3 * 3, 4 * 3) = (-9, 12)
  • Final coordinates: (1 -9, 2 + 12) = P'(-8, 14)

• For Point Q:

  • Displacement from center: (4 -1, -2 -2) = (3, -4)
  • Scaled displacement: (3 * 3, -4 * 3) = (9, -12)
  • Final coordinates: (1 + 9, 2 - 12) = Q'(10, -10)

Therefore, the dilated endpoints are P'(-8, 14) and Q'(10, -10).

Problem 4:

True or False: A dilation always preserves the angles of a polygon.

Answer 4: True. Dilations preserve angle measures; only the size of the figure changes.

Problem 5:

A rectangle with vertices H(0, 0), I(4, 0), J(4, 2), and K(0, 2) is dilated with a center of dilation at (2,1) and a scale factor of 1/2. Find the coordinates of the vertices of the dilated rectangle.

Answer 5:

Again, we need to consider the displacement from the center of dilation (2, 1):

• For Point H:

  • Displacement: (0-2, 0-1) = (-2, -1)
  • Scaled displacement: (-2 * 1/2, -1 * 1/2) = (-1, -0.5)
  • Final coordinates: (2 + (-1), 1 + (-0.5)) = H'(1, 0.5)

• For Point I:

  • Displacement: (4-2, 0-1) = (2, -1)
  • Scaled displacement: (2 * 1/2, -1 * 1/2) = (1, -0.5)
  • Final coordinates: (2 + 1, 1 + (-0.5)) = I'(3, 0.5)

• For Point J:

  • Displacement: (4-2, 2-1) = (2, 1)
  • Scaled displacement: (2 * 1/2, 1 * 1/2) = (1, 0.5)
  • Final coordinates: (2 + 1, 1 + 0.5) = J'(3, 1.5)

• For Point K:

  • Displacement: (0-2, 2-1) = (-2, 1)
  • Scaled displacement: (-2 * 1/2, 1 * 1/2) = (-1, 0.5)
  • Final coordinates: (2 + (-1), 1 + 0.5) = K'(1, 1.5)

The vertices of the dilated rectangle are H'(1, 0.5), I'(3, 0.5), J'(3, 1.5), and K'(1, 1.5).

Advanced Concepts and Applications

While the previous problems focused on basic dilation calculations, more complex problems might involve:

• Dilations with negative scale factors: A negative scale factor results in a dilation and a reflection across the center of dilation.
• Compositions of transformations: A dilation can be combined with other transformations like rotations, reflections, or translations.
• Using dilations to prove geometric theorems: Dilations are a powerful tool for proving theorems related to similarity and proportionality.
• Applications in real-world scenarios: Dilations are used in many fields, including architecture, engineering, and computer graphics (scaling images).

Troubleshooting Common Mistakes

Students often make mistakes when working with dilations. Here are some common errors and how to avoid them:

• Incorrectly applying the scale factor: Double-check your calculations to ensure you are multiplying the coordinates correctly by the scale factor.
• Forgetting the center of dilation: Always remember that the center of dilation remains fixed during the transformation.
• Mixing up x and y coordinates: Pay close attention to which coordinate is being multiplied by the scale factor.
• Incorrectly handling negative scale factors: Remember that a negative scale factor involves both dilation and reflection.

Further Practice and Resources

To further enhance your understanding of dilations, try creating your own practice problems. Start with simple shapes and gradually increase the complexity. Experiment with different centers of dilation and scale factors. You can also search online for additional practice worksheets or interactive exercises focusing on dilations. Remember to always check your answers carefully and seek help if needed.

This comprehensive guide provides a solid foundation for understanding and mastering dilations in geometry. By practicing the problems and understanding the concepts explained here, you'll be well-prepared to tackle more challenging problems and apply this crucial geometric transformation in various contexts. Remember, consistent practice is key to mastering any mathematical concept!