Problem 3 - Algebra 2

Question

Triangle \(A B C\) with vertices \(A(1,4), B(2,-5),\) and \(C(-6,-6)\) is translated 3 units right and 1 unit down. Graph the preimage and the image.

Step-by-Step Solution

Answer

The triangle is translated to new points (4, 3), (5, -6), and (-3, -7).

Step 1: Understand the Translation

A translation involves moving a figure without rotating or reflecting it. In this case, each point of triangle \(ABC\) is moved 3 units to the right and 1 unit down.

Step 2: Apply the Translation to Point A

The original coordinates of point \(A\) are \((1, 4)\). Translate point \(A\) 3 units to the right by adding 3 to the \(x\)-coordinate, and 1 unit down by subtracting 1 from the \(y\)-coordinate: \((1 + 3, 4 - 1) = (4, 3)\).

Step 3: Apply the Translation to Point B

The original coordinates of point \(B\) are \((2, -5)\). Translate point \(B\) 3 units to the right by adding 3 to the \(x\)-coordinate, and 1 unit down by subtracting 1 from the \(y\)-coordinate: \((2 + 3, -5 - 1) = (5, -6)\).

Step 4: Apply the Translation to Point C

The original coordinates of point \(C\) are \((-6, -6)\). Translate point \(C\) 3 units to the right by adding 3 to the \(x\)-coordinate, and 1 unit down by subtracting 1 from the \(y\)-coordinate: \((-6 + 3, -6 - 1) = (-3, -7)\).

Step 5: Graph the Preimage and the Image

Plot the vertices of the original triangle \(ABC\) with points \((1, 4), (2, -5), (-6, -6)\) on a graph. Then plot the translated image with the new points \((4, 3), (5, -6), (-3, -7)\), and draw the triangles to see the translation.

Key Concepts

Translations

Translations in geometry involve moving every point of a figure by the same distance in a given direction. This shift is called a translation vector. In the exercise you're working with, triangle \(ABC\) is translated 3 units to the right and 1 unit down.

A translation keeps the shape and size of the figure intact. There is no rotation or flipping involved; the orientation remains the same.

To translate a point, adjust its coordinates. For example, given a point \( (x, y) \), you adjust its position to \((x + a, y + b)\), where \(a\) and \(b\) are the translations along the x-axis and y-axis, respectively.

• In this case, \(a = 3\) (3 units to the right),
• and \(b = -1\) (1 unit down).

This adjustment shifts all triangle vertices simultaneously, leading to a new triangle on the coordinate plane.

Coordinate Geometry

Coordinate geometry, or Cartesian geometry, utilizes a grid, which is structured by the x-axis and y-axis, to describe locations uniquely by ordered pairs \((x, y)\).

Each point has a unique position determined by how far it is from the origin – (0,0). Positive directions typically extend rightwards along the x-axis and upwards along the y-axis.

When translating geometric figures like triangles, understanding how these axes work is crucial. It allows for precise alterations of points. Observing our exercise with triangle \(ABC\):

• The original position of point \(A\) was \((1, 4)\). Applying the translation vector \((3, -1)\) results in the new location \((4, 3)\).
• Similarly, point \(B\) moves from \((2, -5)\) to \((5, -6)\).
• Finally, point \(C\) shifts from \((-6, -6)\) to \((-3, -7)\).

Mastering coordinate geometry is fundamental for analyzing translations and transformations precisely.

Graphing Triangles

Graphing triangles involves plotting points on the coordinate plane and connecting these to form a polygon. In the exercise, both the original triangle, or preimage, and the translated triangle, or image, need to be graphed.

Start by plotting the original triangle \(ABC\) with given vertices at \((1, 4)\), \((2, -5)\), and \((-6, -6)\). Connect these points linearly to form the triangle.

Next, plot the translated triangle using the new vertices \((4, 3)\), \((5, -6)\), and \((-3, -7)\), obtained from the translation operation.

• Carefully draw lines between each set of points to form congruent triangles.
• Observe that these triangles are the same size and shape, showcasing the uniformity of translation.

Graphing provides a visual representation of the transformation, making abstract geometric concepts tangible and understandable.