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Chapter 9: Problem 32

Graph the image of each figure under a translation by the given vectors. pentagon STUVW with vertices \(S(1,4), T(3,8), U(6,8), V(6,6), W(4,4)\) \(\vec{p}=\langle- 4,5\rangle, \vec{q}=\langle 12,11\rangle\)

Short Answer

Expert verified
Pentagon STUVW translates to new vertices: Under \(\vec{p}: (-3, 9), (-1, 13), (2, 13), (2, 11), (0, 9)\); Under \(\vec{q}: (13, 15), (15, 19), (18, 19), (18, 17), (16, 15)\).

Step by step solution

01

Understand Translation Vectors

Translation involves moving a shape without rotating or resizing it. The vector \(\vec{p}=\langle -4,5 \rangle\) indicates a shift 4 units left and 5 units up, while \(\vec{q}=\langle 12,11 \rangle\) represents a shift 12 units right and 11 units up.
02

Translate Pentagon STUVW by \(\vec{p}\)

Apply the vector \(\vec{p}=\langle -4,5 \rangle\) to each vertex of the pentagon:- For \(S(1,4)\), new coordinates are \((1-4, 4+5) = (-3, 9)\).- For \(T(3,8)\), new coordinates are \((3-4, 8+5) = (-1, 13)\).- For \(U(6,8)\), new coordinates are \((6-4, 8+5) = (2, 13)\).- For \(V(6,6)\), new coordinates are \((6-4, 6+5) = (2, 11)\).- For \(W(4,4)\), new coordinates are \((4-4, 4+5) = (0, 9)\).
03

Translate Pentagon STUVW by \(\vec{q}\)

Now apply the vector \(\vec{q}=\langle 12,11 \rangle\) to each vertex of the pentagon:- For \(S(1,4)\), new coordinates are \((1+12, 4+11) = (13, 15)\).- For \(T(3,8)\), new coordinates are \((3+12, 8+11) = (15, 19)\).- For \(U(6,8)\), new coordinates are \((6+12, 8+11) = (18, 19)\).- For \(V(6,6)\), new coordinates are \((6+12, 6+11) = (18, 17)\).- For \(W(4,4)\), new coordinates are \((4+12, 4+11) = (16, 15)\).
04

Plot the Translated Figures

On graph paper or using graphing software, plot the translated coordinates from Step 2 to form the image of the pentagon after translation by \(\vec{p}\). Then, plot the translated coordinates from Step 3 to form the image of the pentagon after translation by \(\vec{q}\). Connect the vertices in the order listed to complete each pentagon.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Transformations
In geometry, transformation refers to moving a shape or figure in the plane. Among the transformations are translations, which involve shifting every point in a figure the same distance and direction across the plane. Key properties are:
  • No rotation or resizing occurs鈥攖he shape remains congruent.
  • The orientation remains the same.
  • Only the position of the shape changes.
When you apply a vector, like \( \vec{p} = \langle -4,5 \rangle \), each vertex of a figure moves accordingly 鈥� 4 units left (negative x-direction) and 5 units up (positive y-direction). This consistency ensures the entire shape translates smoothly.
Basics of Coordinate Geometry
Coordinate geometry, also termed analytic geometry, involves plotting points, lines, and shapes on a graph using coordinates. The Cartesian coordinate system is most commonly used, with two perpendicular number lines called axes: the x-axis (horizontal) and the y-axis (vertical).
  • Each point is identified by an ordered pair (x, y).
  • The origin is where both axes intersect, denoted as (0,0).
  • Positive and negative values on the axes determine the quadrants.
Understanding coordinate geometry is crucial for translating shapes. By taking an ordered pair and modifying it by a vector's movement instructions, you can accurately reposition each vertex of a shape, as depicted in the translation steps for pentagon STUVW.
Graphing Figures After Transformation
Graphing figures after a transformation requires careful plotting of the new coordinates influenced by the translation vectors. Here are some steps to ensure accuracy:
  • Apply the translation vector to all original vertices to get a new set of coordinates.
  • Plot each new point on the graph according to its x and y values.
  • Repeatedly check each point's position against the graph鈥檚 scale.
  • Connect the new dots in the same order as the original figure to preserve the shape.
For the pentagon exercise, after applying vectors \( \vec{p} \) and \( \vec{q} \), graph each translated set of coordinates, ensuring the original sequence from S to W is maintained, resulting in the transformed and accurately graphed figures.

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