Congruence Transformations

Describe the transformation that maps triangle ABC onto A'B'C'.

Transformation mapping triangle ABC to A'B'C' - StudySmarter Originals

Solution

From the figure, we can see that triangle ABC has simply been moved to another location, resulting in triangle A'B'C'. When describing translations, we use vectors. In the above diagram, if we look at point A, we can see it has been moved three spaces to the right and four spaces up to get to the point A'. This is the same for point B and C. Thus, we can say that the shape has been translated by the vector $\left(\begin{array}{c}3\\ 4\end{array}\right)$.

Translate the below quadrilateral by the vector $\left(\begin{array}{c}-4\\ -3\end{array}\right)$.

Translation Example - StudySmarter Originals

Solution

To translate the shape, it helps to look at each point individually. If we take point A, we need to move it 4 spaces to the left and 3 spaces down to get to the point A'. If we do the same for points B, C, and D, we get the below quadrilateral A'B'C'D'.

Translation Example - StudySmarter Originals

Reflect the shape ABCD across the y-axis.

Reflection Example 1 - StudySmarter Originals

Solution

Since we are reflecting across the y-axis, the shape will only move horizontally and only the x-values will change. To reflect across the y-axis, we multiply the x-values by -1, like so:

• Point A (2,5) is reflected to point A' (-2,5)
• Point B (4,5) is reflected to point B' (-4,5)
• Point C (4,3) is reflected to point C' (-4,3)
• Point D (2,3) is reflected to point D' (-2,3)

See the reflected points A', B', C', and D' plotted in the graph below.

Reflection Example 1 - StudySmarter Originals

In the shape below, describe the single transformation that maps the triangle ABC to triangle A'B'C'.

Reflection Example 2 - StudySmarter Originals

Solution

If we look closely at the figure above, we can see that the shape ABC has been reflected over the line $y=x$. Therefore, we have a reflection over the line $y=x$.

This line has a positive slope of 1, and it passes directly through the origin (0,0). To reflect over the line y=x, we swap the points' x- and y-coordinates, like so:

• Point A (3,-1) is reflected to point A' (-1,3)
• Point B (3,-3) is reflected to point B' (-3,3)
• Point C (1,-3) is reflected to point C' (-3,1)

Describe the single transformation that maps shape ABCDE onto A'B'C'D'E'.

Rotation Example 1 - StudySmarter Originals

Solution

We can tell by looking at the transformed shape that there has been a rotation of 90 degrees counterclockwise. However, we need to determine the center of rotation. If we look at the points E and E', we can see that a right angle is formed about the origin. Therefore, the origin is the center of rotation for this shape's congruence transformation. Thus, we have a rotation of 90 degrees counterclockwise about the origin.

To rotate an object 90 degrees about the origin, we perform the following adjustment to our coordinates:

(x,y) becomes (-y,x), like so:

• Point A (2,3) is rotated to point A' (-3,2)
• Point B (4,3) is rotated to point B' (-3,4)
• Point C (4,0) is rotated to point C' (0,4)
• Point D (3,1) is rotated to point D' (-1,3)
• Point E (2,0) is rotated to point E' (0,2)

Rotate the triangle ABC 180 degrees about the origin.

Rotation Example 2 - StudySmarter Originals

Solution

If we consider point A's distance from the origin, we can see that it is 1 unit to the right and 3 units up. If we rotate it 180 degrees clockwise, it will be 1 unit to the left of the origin and three units down. The same applies to points B and C. We have the below rotation.

Rotation Example 2 - StudySmarter Originals

To rotate an object 180 degrees about the origin, we perform the following adjustment to our coordinates:

(x,y) becomes (-x,-y), like so:

• Point A (1,3) is rotated to point A' (-1,-3)
• Point B (3,1) is rotated to point B' (-3,-1)
• Point C (4,4) is rotated to point C' (-4,-4)