Question

U4 L9 Question 2
Video Player

Question
Using triangle ABC with coordinates A(5,3) B(-2,7) C(-3,-8). Rotate the triangle 270 degrees counter-clockwise (about the origin) and then reflect across the line y=4. What are the coordinates of A''?(1 point)
Responses

(5,7)
(5,7)

(-6,-8)
(-6,-8)

(10,-4)
(10,-4)

(3,13)

Answer 1

To determine the coordinates of point A'' after a rotation of 270 degrees counter-clockwise about the origin and a reflection across the line \(y=4\), we can follow these steps:

1. **Rotate point A(5,3) by 270 degrees counter-clockwise about the origin**:
The rotation formula for a point \((x, y)\) counter-clockwise by an angle \(\theta\) is given by:
\[
(x', y') = (x \cdot \cos(\theta) - y \cdot \sin(\theta), x \cdot \sin(\theta) + y \cdot \cos(\theta))
\]
For a rotation of 270 degrees (which is equivalent to -90 degrees), the cosine and sine values are:
\[
\cos(270^\circ) = 0, \quad \sin(270^\circ) = -1
\]

Plugging in \(A(5,3)\):
\[
A' = (5 \cdot 0 - 3 \cdot -1, 5 \cdot -1 + 3 \cdot 0) = (0 + 3, -5 + 0) = (3, -5)
\]

2. **Reflect point A'(3,-5) across the line \(y=4\)**:
To reflect a point \((x, y)\) across a horizontal line \(y=k\), the formula is:
\[
(x, y') = (x, k - (y - k)) = (x, 2k - y)
\]

Here, \(k = 4\):
\[
A'' = (3, 2 \cdot 4 - (-5)) = (3, 8 + 5) = (3, 13)
\]

Thus, the coordinates of point A'' are \((3, 13)\).

The correct response is:
**(3, 13)**.