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Triangle JKL

has vertices J(3, –7)
, K(7, –3)
and L(3, 7)
. What are the coordinates of J
after a 90°
rotation counterclockwise about the origin? Enter the answer.

5 answers

To perform a 90° counterclockwise rotation about the origin, you can use the transformation:

\[ (x, y) \rightarrow (-y, x) \]

Starting with the coordinates of point J, which are \( J(3, -7) \):

  1. Take the \( y \)-coordinate: \(-7\)
  2. Take the \( x \)-coordinate: \(3\)

Now apply the transformation:

\[ J'(x', y') = (-(-7), 3) = (7, 3) \]

So the coordinates of point J after a 90° rotation counterclockwise about the origin are:

\[ \boxed{(7, 3)} \]

Figure ABCD
has vertices A(–2, –2)
, B(1, 1)
, C(0, 4)
, and D(–3, 5)
. The figure is translated 5
units to the right and 6
units down. Then it is translated 2
units to the left and 2
units up. What are the coordinates of image A″B″C″D″
after both translations? Enter the answers.

Let's perform the translations step-by-step for each vertex of the figure ABCD.

Step 1: Initial Coordinates

  • A: (−2, −2)
  • B: (1, 1)
  • C: (0, 4)
  • D: (−3, 5)

Step 2: First Translation

The first translation is 5 units to the right and 6 units down. This means we add 5 to the x-coordinate and subtract 6 from the y-coordinate.

Translations:

  1. A(−2, −2): \[ A' = (−2 + 5, −2 - 6) = (3, -8) \]
  2. B(1, 1): \[ B' = (1 + 5, 1 - 6) = (6, -5) \]
  3. C(0, 4): \[ C' = (0 + 5, 4 - 6) = (5, -2) \]
  4. D(−3, 5): \[ D' = (−3 + 5, 5 - 6) = (2, -1) \]

Step 3: Second Translation

The second translation is 2 units to the left and 2 units up. This means we subtract 2 from the x-coordinate and add 2 to the y-coordinate.

Translations:

  1. A'(3, −8): \[ A'' = (3 - 2, -8 + 2) = (1, -6) \]
  2. B'(6, −5): \[ B'' = (6 - 2, -5 + 2) = (4, -3) \]
  3. C'(5, −2): \[ C'' = (5 - 2, -2 + 2) = (3, 0) \]
  4. D'(2, −1): \[ D'' = (2 - 2, -1 + 2) = (0, 1) \]

Final Coordinates of A'', B'', C'', and D''

The coordinates of the image \( A''B''C''D'' \) after the two translations are:

  • A'': \( (1, -6) \)
  • B'': \( (4, -3) \)
  • C'': \( (3, 0) \)
  • D'': \( (0, 1) \)

So, the answers are:

  • A'' = \( (1, -6) \)
  • B'' = \( (4, -3) \)
  • C'' = \( (3, 0) \)
  • D'' = \( (0, 1) \)

You can represent this as:

  • A'' = \( \boxed{(1, -6)} \)
  • B'' = \( \boxed{(4, -3)} \)
  • C'' = \( \boxed{(3, 0)} \)
  • D'' = \( \boxed{(0, 1)} \)
Triangle ABC has coordinates A(3, 3) , B(5, –2) , and C(4, 4) . Graph the preimage and image of the triangle after it is reflected across the line x=1 .
Explain how to write the coordinates of the vertices of the reflected triangle. Enter the answer.

To reflect the triangle ABC across the line \( x = 1 \), we need to determine how the coordinates of each vertex transform under this reflection.

Reflection Rule:

When reflecting a point \((x, y)\) across the vertical line \(x = k\), the new x-coordinate can be calculated using the formula: \[ x' = 2k - x \] The y-coordinate remains unchanged: \[ y' = y \]

In this case, the line is \(x = 1\). Therefore, for each vertex \((x, y)\), the reflected coordinates will be: \[ x' = 2(1) - x = 2 - x \] \[ y' = y \]

Step 1: Reflect each vertex of the triangle ABC

  1. Vertex A(3, 3):

    • \( x' = 2 - 3 = -1 \)
    • \( y' = 3 \)
    • Thus, \( A' = (-1, 3) \)

  2. Vertex B(5, -2):

    • \( x' = 2 - 5 = -3 \)
    • \( y' = -2 \)
    • Thus, \( B' = (-3, -2) \)

  3. Vertex C(4, 4):

    • \( x' = 2 - 4 = -2 \)
    • \( y' = 4 \)
    • Thus, \( C' = (-2, 4) \)

Step 2: Summarize the coordinates of the reflected triangle

The coordinates of the vertices of the reflected triangle \( A'B'C' \) are:

  • \( A' = (-1, 3) \)
  • \( B' = (-3, -2) \)
  • \( C' = (-2, 4) \)

Result:

So, the coordinates of the vertices of the reflected triangle are:

  • \( A' = \boxed{(-1, 3)} \)
  • \( B' = \boxed{(-3, -2)} \)
  • \( C' = \boxed{(-2, 4)} \)

Explanation:

To reflect the vertices of triangle ABC across the line \( x = 1 \), we determined the new x-coordinates by applying the reflection formula while keeping the y-coordinates unchanged. The resulting coordinates represent the vertices of the reflected triangle \( A'B'C' \).