3. The diameter of a circle has endpoints at (11,7) and (-13,3). What are the coordinates of the center of the circle?

Holt Mcdougal Larson Pre-algebra: Student Edition 2012
1st Edition
ISBN:9780547587776
Author:HOLT MCDOUGAL
Publisher:HOLT MCDOUGAL
Chapter9: Real Numbers And Right Triangles
Section9.5: The Distance And Midpoint Formulas
Problem 2C
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### Question 3:

**Problem Statement:**
The diameter of a circle has endpoints at \( (11, 7) \) and \( (-13, 3) \). What are the coordinates of the center of the circle?

---

**Solution Explanation:**

To find the center of the circle, we need to find the midpoint of the line segment with endpoints at \( (11, 7) \) and \( (-13, 3) \). The formula for the midpoint, \( M \), of a line segment with endpoints \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:

\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

Plugging in the given coordinates:

\[ 
\begin{align*}
x_1 &= 11, & y_1 &= 7 \\
x_2 &= -13, & y_2 &= 3 
\end{align*}
\]

**Coordinates Calculation:**

\[ 
\begin{align*}
M_x &= \frac{11 + (-13)}{2} = \frac{11 - 13}{2} = \frac{-2}{2} = -1, \\
M_y &= \frac{7 + 3}{2} = \frac{10}{2} = 5 
\end{align*}
\]

Thus, the coordinates of the center of the circle are \( (-1, 5) \).

### Answer:
The coordinates of the center of the circle are \( (-1, 5) \).

---

**Graph Explanation:**
There is a blank rectangular placeholder in the lower part of the image. This space can be utilized for any additional diagrams or graphs to visually illustrate the solution, such as a diagram showing the circle and its diameter with marked endpoints and the center.

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Transcribed Image Text:--- ### Question 3: **Problem Statement:** The diameter of a circle has endpoints at \( (11, 7) \) and \( (-13, 3) \). What are the coordinates of the center of the circle? --- **Solution Explanation:** To find the center of the circle, we need to find the midpoint of the line segment with endpoints at \( (11, 7) \) and \( (-13, 3) \). The formula for the midpoint, \( M \), of a line segment with endpoints \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \] Plugging in the given coordinates: \[ \begin{align*} x_1 &= 11, & y_1 &= 7 \\ x_2 &= -13, & y_2 &= 3 \end{align*} \] **Coordinates Calculation:** \[ \begin{align*} M_x &= \frac{11 + (-13)}{2} = \frac{11 - 13}{2} = \frac{-2}{2} = -1, \\ M_y &= \frac{7 + 3}{2} = \frac{10}{2} = 5 \end{align*} \] Thus, the coordinates of the center of the circle are \( (-1, 5) \). ### Answer: The coordinates of the center of the circle are \( (-1, 5) \). --- **Graph Explanation:** There is a blank rectangular placeholder in the lower part of the image. This space can be utilized for any additional diagrams or graphs to visually illustrate the solution, such as a diagram showing the circle and its diameter with marked endpoints and the center. ---
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