Right Triangle { LMN $}$ Has Vertices { L(7,-3), M(7,-8), $}$ And { N(10,-8) $}$. The Triangle Is Translated On The Coordinate Plane So That The Coordinates Of { L^{\prime} $}$ Are { (-1,8)$}$.Which

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Introduction

In mathematics, the concept of translation is a fundamental transformation that involves shifting a figure from one location to another on the coordinate plane. This transformation does not change the size or shape of the figure, but rather its position. In this article, we will explore the concept of translation in the context of a right triangle, specifically triangle LMN. We will examine the vertices of the triangle, its translation, and the resulting coordinates of the translated triangle.

The Original Triangle LMN

The original triangle LMN has vertices L(7, -3), M(7, -8), and N(10, -8). To understand the concept of translation, let's first visualize the triangle on the coordinate plane.

  L(7, -3)
 /   \
M(7, -8)
 \   /
  N(10, -8)

Translation of Triangle LMN

The triangle LMN is translated on the coordinate plane so that the coordinates of L' are (-1, 8). This means that the triangle is shifted horizontally and vertically from its original position to a new position.

  L'(-1, 8)
 /   \
M'(x, y)
 \   /
  N'(x, y)

Determining the Coordinates of M' and N'

To determine the coordinates of M' and N', we need to understand the concept of translation. When a figure is translated, each point on the figure is shifted by the same distance and direction. In this case, the triangle is shifted 8 units up and 8 units left from its original position.

  L(7, -3) -> L'(-1, 8) (shifted 8 units up and 8 units left)

Using this information, we can determine the coordinates of M' and N'. Since the triangle is shifted 8 units up and 8 units left, the x-coordinate of M' and N' will be 8 units less than the x-coordinate of M and N, respectively. The y-coordinate of M' and N' will be 8 units more than the y-coordinate of M and N, respectively.

  M(7, -8) -> M'(-1 + 8, -8 + 8) = M'(7, 0)
  N(10, -8) -> N'(-1 + 8, -8 + 8) = N'(7, 0)

Conclusion

In conclusion, the triangle LMN is translated on the coordinate plane so that the coordinates of L' are (-1, 8). By understanding the concept of translation, we can determine the coordinates of M' and N'. The resulting triangle has vertices L'(-1, 8), M'(7, 0), and N'(7, 0).

Key Takeaways

  • The concept of translation involves shifting a figure from one location to another on the coordinate plane.
  • The size and shape of the figure remain unchanged during translation.
  • Each point on the figure is shifted by the same distance and direction.
  • The coordinates of the translated figure can be determined using the concept of translation.

Real-World Applications

The concept of translation has numerous real-world applications in fields such as engineering, architecture, and computer graphics. It is used to create 3D models, animations, and special effects in movies and video games.

Practice Problems

  1. A triangle has vertices A(2, 3), B(4, 5), and C(6, 7). If the triangle is translated 3 units up and 2 units left, what are the coordinates of the translated triangle?
  2. A rectangle has vertices D(1, 2), E(3, 4), F(5, 6), and G(7, 8). If the rectangle is translated 2 units down and 1 unit right, what are the coordinates of the translated rectangle?

Solutions

  1. The translated triangle has vertices A'(-2 + 3, 3 + 3) = A'(1, 6), B'(-2 + 3, 5 + 3) = B'(1, 8), and C'(-2 + 3, 7 + 3) = C'(1, 10).
  2. The translated rectangle has vertices D'(1 - 1, 2 - 2) = D'(0, 0), E'(3 - 1, 4 - 2) = E'(2, 2), F'(5 - 1, 6 - 2) = F'(4, 4), and G'(7 - 1, 8 - 2) = G'(6, 6).

Conclusion

Frequently Asked Questions

Q1: What is translation in the context of right triangle LMN?

A1: Translation is a fundamental transformation that involves shifting a figure from one location to another on the coordinate plane. In the context of right triangle LMN, translation means moving the triangle from its original position to a new position.

Q2: How do I determine the coordinates of the translated triangle?

A2: To determine the coordinates of the translated triangle, you need to understand the concept of translation. When a figure is translated, each point on the figure is shifted by the same distance and direction. In this case, the triangle is shifted 8 units up and 8 units left from its original position.

Q3: What are the coordinates of the translated triangle?

A3: The coordinates of the translated triangle are L'(-1, 8), M'(7, 0), and N'(7, 0).

Q4: How do I apply the concept of translation to real-world problems?

A4: The concept of translation has numerous real-world applications in fields such as engineering, architecture, and computer graphics. It is used to create 3D models, animations, and special effects in movies and video games.

Q5: What are some common mistakes to avoid when working with translation?

A5: Some common mistakes to avoid when working with translation include:

  • Failing to understand the concept of translation and its application to real-world problems.
  • Not considering the direction and distance of the translation.
  • Not using the correct coordinates of the translated figure.

Q6: How do I practice working with translation?

A6: To practice working with translation, try the following exercises:

  • Translate a figure on the coordinate plane by a given distance and direction.
  • Determine the coordinates of the translated figure.
  • Apply the concept of translation to real-world problems.

Q7: What are some advanced topics related to translation?

A7: Some advanced topics related to translation include:

  • Reflection: a transformation that involves flipping a figure over a line or axis.
  • Rotation: a transformation that involves rotating a figure around a point or axis.
  • Scaling: a transformation that involves changing the size of a figure.

Q8: How do I use translation in computer graphics?

A8: Translation is used extensively in computer graphics to create 3D models, animations, and special effects in movies and video games. It is also used to create interactive simulations and visualizations.

Q9: What are some real-world applications of translation?

A9: Some real-world applications of translation include:

  • Engineering: translation is used to design and build complex structures such as bridges and buildings.
  • Architecture: translation is used to design and build complex buildings and structures.
  • Computer graphics: translation is used to create 3D models, animations, and special effects in movies and video games.

Q10: How do I use translation in mathematics?

A10: Translation is used extensively in mathematics to solve problems involving geometry and trigonometry. It is also used to create mathematical models and simulations.

Conclusion

In conclusion, the concept of translation is a fundamental transformation that involves shifting a figure from one location to another on the coordinate plane. By understanding the concept of translation, we can determine the coordinates of the translated figure and apply it to real-world problems.