Consider the figure below.
Triangle P-Q-R has vertex P at (-5, 2), Q at (-2, 7), R at (-2, 2) and triangle P prime Q prime R prime has vertex P prime at (4, -2), Q prime at (7, -7), R prime at (7, -2)
Determine if the statements in the table below are true or false about the sequences of transformations that can verify that is congruent to.
Drag and drop either True or False into the box next to each statement.
Statement True or False
Triangle PQR is translated 9 units down and then translated 9 units to the right.
Triangle PQR is reflected across the x-axis and then translated 9 units to the right.
Triangle PQR is rotated 180° clockwise about the origin and then translated 2 units to the right.
False, False, True
To determine if the statements are true or false, we need to analyze the given figure and the sequences of transformations mentioned.
Statement 1: Triangle PQR is translated 9 units down and then translated 9 units to the right.
To verify this statement, let's apply the translations to triangle PQR:
- First, a translation of 9 units down would mean adding 9 to the y-coordinate of each vertex. P'Q'R' would have vertex P' at (-5, 2 + 9) = (-5, 11), vertex Q' at (-2, 7 + 9) = (-2, 16), and vertex R' at (-2, 2 + 9) = (-2, 11).
- Next, a translation of 9 units to the right would mean adding 9 to the x-coordinate of each vertex. However, if we apply this translation, triangle P'Q'R' would have vertex P' at (-5 + 9, 11) = (4, 11), which does not match the given coordinates for P prime. Therefore, this statement is False.
Statement 2: Triangle PQR is reflected across the x-axis and then translated 9 units to the right.
To verify this statement, let's apply the transformations to triangle PQR:
- Reflection across the x-axis would mean changing the sign of the y-coordinate for each vertex. Triangle P'Q'R' would have vertex P' at (-5, -2), vertex Q' at (-2, -7), and vertex R' at (-2, -2).
- Now, a translation of 9 units to the right would mean adding 9 to the x-coordinate of each vertex. P'Q'R' would have vertex P' at (-5 + 9, -2) = (4, -2), vertex Q' at (-2 + 9, -7) = (7, -7), and vertex R' at (-2 + 9, -2) = (7, -2). These coordinates match the given coordinates for P prime, Q prime, and R prime. Therefore, this statement is True.
Statement 3: Triangle PQR is rotated 180° clockwise about the origin and then translated 2 units to the right.
To verify this statement, let's apply the transformations to triangle PQR:
- A rotation of 180° clockwise about the origin would mean negating both the x-coordinate and the y-coordinate of each vertex. Triangle P'Q'R' would have vertex P' at (5, -2), vertex Q' at (2, -7), and vertex R' at (2, -2).
- Next, a translation of 2 units to the right would mean adding 2 to the x-coordinate of each vertex. P'Q'R' would have vertex P' at (5 + 2, -2) = (7, -2), vertex Q' at (2 + 2, -7) = (4, -7), and vertex R' at (2 + 2, -2) = (4, -2). These coordinates do not match the given coordinates for P prime, Q prime, and R prime. Therefore, this statement is False.
In summary:
- Statement 1 is False.
- Statement 2 is True.
- Statement 3 is False.
To determine if the statements are true or false, let's analyze each transformation:
1. Triangle PQR is translated 9 units down and then translated 9 units to the right.
To perform this transformation on triangle PQR, we would first move each point 9 units down and then 9 units to the right. Let's apply these translations to the coordinates of triangle PQR:
Point P (-5, 2) -> (-5 + 9, 2 - 9) = (4, -7)
Point Q (-2, 7) -> (-2 + 9, 7 - 9) = (7, -2)
Point R (-2, 2) -> (-2 + 9, 2 - 9) = (7, -7)
Comparing these new coordinates with the coordinates of triangle P'Q'R', we can see that they do not match. Therefore, the first statement is False.
2. Triangle PQR is reflected across the x-axis and then translated 9 units to the right.
If we reflect triangle PQR across the x-axis, the y-coordinates will change their signs. Let's perform this reflection:
Point P (-5, 2) -> (-5, -2)
Point Q (-2, 7) -> (-2, -7)
Point R (-2, 2) -> (-2, -2)
Now, if we apply the translation of 9 units to the right:
Point P (-5, -2) -> (-5 + 9, -2) = (4, -2)
Point Q (-2, -7) -> (-2 + 9, -7) = (7, -7)
Point R (-2, -2) -> (-2 + 9, -2) = (7, -2)
Comparing these new coordinates with the coordinates of triangle P'Q'R', we can see that they match. Therefore, the second statement is True.
3. Triangle PQR is rotated 180° clockwise about the origin and then translated 2 units to the right.
To rotate triangle PQR 180° clockwise about the origin, we need to change the sign of both the x and y-coordinates. Let's perform this rotation:
Point P (-5, 2) -> (5, -2)
Point Q (-2, 7) -> (2, -7)
Point R (-2, 2) -> (2, -2)
Now, if we apply the translation of 2 units to the right:
Point P (5, -2) -> (5 + 2, -2) = (7, -2)
Point Q (2, -7) -> (2 + 2, -7) = (4, -7)
Point R (2, -2) -> (2 + 2, -2) = (4, -2)
Comparing these new coordinates with the coordinates of triangle P'Q'R', we can see that they match. Therefore, the third statement is True.
In summary:
Statement 1: False
Statement 2: True
Statement 3: True