Triangle PQR Has Vertices \[$ P(-2, 6) \$\], \[$ Q(-8, 4) \$\], And \[$ R(1, -2) \$\]. It Is Translated According To The Rule \[$(x, Y) \rightarrow (x-2, Y-16)\$\].What Is The \[$ Y \$\]-value Of \[$ P'

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Triangle Translation: Finding the New Coordinates of P'

In geometry, translation is a fundamental concept that involves moving a point or a shape from one location to another without changing its size or orientation. In this article, we will explore the concept of translation and apply it to a given triangle PQR. We will use the translation rule (x,y)β†’(xβˆ’2,yβˆ’16){(x, y) \rightarrow (x-2, y-16)} to find the new coordinates of point P'.

Understanding the Translation Rule

The translation rule (x,y)β†’(xβˆ’2,yβˆ’16){(x, y) \rightarrow (x-2, y-16)} indicates that for every point (x, y), we need to subtract 2 from the x-coordinate and subtract 16 from the y-coordinate to get the new coordinates. This rule will be applied to each vertex of the triangle PQR to find the new coordinates.

Finding the New Coordinates of P'

The original coordinates of point P are (-2, 6). To find the new coordinates of P', we will apply the translation rule:

(x,y)β†’(xβˆ’2,yβˆ’16){(x, y) \rightarrow (x-2, y-16)}

Substituting the original coordinates of P into the translation rule, we get:

(x,y)=(βˆ’2,6){(x, y) = (-2, 6)} (xβˆ’2,yβˆ’16)=(βˆ’2βˆ’2,6βˆ’16){(x-2, y-16) = (-2-2, 6-16)} (xβˆ’2,yβˆ’16)=(βˆ’4,βˆ’10){(x-2, y-16) = (-4, -10)}

Therefore, the new coordinates of P' are (-4, -10).

Finding the New Coordinates of Q'

The original coordinates of point Q are (-8, 4). To find the new coordinates of Q', we will apply the translation rule:

(x,y)β†’(xβˆ’2,yβˆ’16){(x, y) \rightarrow (x-2, y-16)}

Substituting the original coordinates of Q into the translation rule, we get:

(x,y)=(βˆ’8,4){(x, y) = (-8, 4)} (xβˆ’2,yβˆ’16)=(βˆ’8βˆ’2,4βˆ’16){(x-2, y-16) = (-8-2, 4-16)} (xβˆ’2,yβˆ’16)=(βˆ’10,βˆ’12){(x-2, y-16) = (-10, -12)}

Therefore, the new coordinates of Q' are (-10, -12).

Finding the New Coordinates of R'

The original coordinates of point R are (1, -2). To find the new coordinates of R', we will apply the translation rule:

(x,y)β†’(xβˆ’2,yβˆ’16){(x, y) \rightarrow (x-2, y-16)}

Substituting the original coordinates of R into the translation rule, we get:

(x,y)=(1,βˆ’2){(x, y) = (1, -2)} (xβˆ’2,yβˆ’16)=(1βˆ’2,βˆ’2βˆ’16){(x-2, y-16) = (1-2, -2-16)} (xβˆ’2,yβˆ’16)=(βˆ’1,βˆ’18){(x-2, y-16) = (-1, -18)}

Therefore, the new coordinates of R' are (-1, -18).

In this article, we applied the translation rule (x,y)β†’(xβˆ’2,yβˆ’16){(x, y) \rightarrow (x-2, y-16)} to find the new coordinates of the vertices of triangle PQR. We found that the new coordinates of P' are (-4, -10), the new coordinates of Q' are (-10, -12), and the new coordinates of R' are (-1, -18). This demonstrates the concept of translation in geometry and how it can be applied to find the new coordinates of points and shapes.

The y-value of P' is -10.

Translation is a fundamental concept in geometry that involves moving a point or a shape from one location to another without changing its size or orientation. It is used to describe the movement of objects in a two-dimensional or three-dimensional space. Translation is an essential concept in geometry, and it has numerous applications in various fields, including art, architecture, engineering, and computer science.

There are several types of transformations in geometry, including:

  • Translation: Moving a point or a shape from one location to another without changing its size or orientation.
  • Rotation: Rotating a point or a shape around a fixed point or axis without changing its size or orientation.
  • Reflection: Reflecting a point or a shape across a line or a plane without changing its size or orientation.
  • Dilation: Enlarging or reducing a point or a shape by a scale factor without changing its orientation.

Translation has numerous applications in real-life scenarios, including:

  • Art and Design: Translation is used to create symmetries and patterns in art and design.
  • Architecture: Translation is used to design and build buildings and structures.
  • Engineering: Translation is used to design and build machines and mechanisms.
  • Computer Science: Translation is used in computer graphics and game development.

Learning translation in geometry has numerous benefits, including:

  • Improved problem-solving skills: Translation helps to develop problem-solving skills and critical thinking.
  • Enhanced spatial reasoning: Translation helps to develop spatial reasoning and visualization skills.
  • Better understanding of geometry: Translation helps to develop a deeper understanding of geometry and its applications.
  • Improved math skills: Translation helps to develop math skills and problem-solving abilities.
    Triangle Translation: Q&A

A: The y-value of P' is -10.

A: Translation is a fundamental concept in geometry that involves moving a point or a shape from one location to another without changing its size or orientation. It is used to describe the movement of objects in a two-dimensional or three-dimensional space. Translation is an essential concept in geometry, and it has numerous applications in various fields, including art, architecture, engineering, and computer science.

A: There are several types of transformations in geometry, including:

  • Translation: Moving a point or a shape from one location to another without changing its size or orientation.
  • Rotation: Rotating a point or a shape around a fixed point or axis without changing its size or orientation.
  • Reflection: Reflecting a point or a shape across a line or a plane without changing its size or orientation.
  • Dilation: Enlarging or reducing a point or a shape by a scale factor without changing its orientation.

A: Translation has numerous applications in real-life scenarios, including:

  • Art and Design: Translation is used to create symmetries and patterns in art and design.
  • Architecture: Translation is used to design and build buildings and structures.
  • Engineering: Translation is used to design and build machines and mechanisms.
  • Computer Science: Translation is used in computer graphics and game development.

A: Learning translation in geometry has numerous benefits, including:

  • Improved problem-solving skills: Translation helps to develop problem-solving skills and critical thinking.
  • Enhanced spatial reasoning: Translation helps to develop spatial reasoning and visualization skills.
  • Better understanding of geometry: Translation helps to develop a deeper understanding of geometry and its applications.
  • Improved math skills: Translation helps to develop math skills and problem-solving abilities.

A: To apply the translation rule, you need to subtract the translation value from the x-coordinate and subtract the translation value from the y-coordinate. For example, if the translation rule is (x,y)β†’(xβˆ’2,yβˆ’16){(x, y) \rightarrow (x-2, y-16)}, you would subtract 2 from the x-coordinate and subtract 16 from the y-coordinate.

A: Translation and rotation are two different types of transformations in geometry. Translation involves moving a point or a shape from one location to another without changing its size or orientation, while rotation involves rotating a point or a shape around a fixed point or axis without changing its size or orientation.

A: Yes, translation is used in the design of buildings and structures. Architects use translation to move a building design from one location to another without changing its size or orientation. For example, if a building design is created in a computer-aided design (CAD) software, the architect can use translation to move the design from one location to another without changing its size or orientation.

A: The translation value in a translation rule is determined by the amount of movement in the x and y directions. For example, if the translation rule is (x,y)β†’(xβˆ’2,yβˆ’16){(x, y) \rightarrow (x-2, y-16)}, the translation value is -2 in the x direction and -16 in the y direction.

A: Translation is a way of moving a point or a shape from one location to another without changing its size or orientation. It's like moving a piece of paper from one place to another without changing its shape or size.