$$ X Y $$ has endpoints $$ X(-3,1) $$ and $$ Y(4,-5) $$. What is the midpoint of its image after a $$ 90^{\circ} $$ rotation about the origin?

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UpStudy AI · Accepted Answer

To find the midpoint of the image of segment $$ XY $$ after a $$ 90^{\circ} $$ rotation about the origin, follow these steps:

1. **Rotate Each Endpoint:**
   - A $$ 90^{\circ} $$ rotation of a point $$(a, b)$$ about the origin results in $$(-b, a)$$.
   - **Point $$ X(-3, 1) $$ rotated:**
     $$
     (-1, -3)
     $$
   - **Point $$ Y(4, -5) $$ rotated:**
     $$
     (5, 4)
     $$

2. **Find the Midpoint of the Rotated Endpoints:**
   - The midpoint $$ M $$ of two points $$(x_1, y_1)$$ and $$(x_2, y_2)$$ is given by:
     $$
     M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
     $$
   - Applying this to the rotated points:
     $$
     M = \left( \frac{-1 + 5}{2}, \frac{-3 + 4}{2} \right) = \left( \frac{4}{2}, \frac{1}{2} \right) = (2, \frac{1}{2})
     $$

**Final Answer:**
$$(2,\ \tfrac{1}{2})$$
The midpoint after the rotation is $$(2, \frac{1}{2})$$.

has endpoints and . What is the midpoint of its image after
a rotation about the origin?

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has endpoints and . What is the midpoint of its image after a rotation about the origin?