1. Calculate the length of $$ C D $$ if $$ C(-9 ; 1) $$ and $$ D(-3 ; 5) $$. 2f the length of $$ A B $$ is 5 , calculate the missing coordinate if $$ A(6 ;-2) $$ and $$ B(x ; 2) $$. 3. Calculate the midpoint of line segment $$ E F $$ if $$ E(-4 ; 7) $$ and $$ F(2 ; 5) $$.

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Sure, let's tackle each problem step by step.

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### **1. Calculate the length of $$ CD $$ where $$ C(-9, 1) $$ and $$ D(-3, 5) $$.**

**Solution:**

To find the distance between two points $$ C(x_1, y_1) $$ and $$ D(x_2, y_2) $$ in a coordinate plane, we use the **distance formula**:

$$
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
$$

**Applying the formula:**

$$
x_1 = -9,\ y_1 = 1 \
x_2 = -3,\ y_2 = 5
$$

$$
\text{Distance} = \sqrt{(-3 - (-9))^2 + (5 - 1)^2} \
= \sqrt{(6)^2 + (4)^2} \
= \sqrt{36 + 16} \
= \sqrt{52} \
= 2\sqrt{13} \approx 7.21 \text{ units}
$$

**Answer:** The length of $$ CD $$ is $$ 2\sqrt{13} $$ units, approximately **7.21 units**.

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### **2. If the length of $$ AB $$ is 5, find the missing coordinate $$ x $$ given $$ A(6, -2) $$ and $$ B(x, 2) $$.**

**Solution:**

Given points $$ A(6, -2) $$ and $$ B(x, 2) $$, and the distance $$ AB = 5 $$.

Again, using the **distance formula**:

$$
\text{Distance} = \sqrt{(x - 6)^2 + (2 - (-2))^2} = 5
$$

Simplify the equation:

$$
\sqrt{(x - 6)^2 + (4)^2} = 5 \
\sqrt{(x - 6)^2 + 16} = 5
$$

Square both sides to eliminate the square root:

$$
(x - 6)^2 + 16 = 25 \
(x - 6)^2 = 25 - 16 \
(x - 6)^2 = 9
$$

Take the square root of both sides:

$$
x - 6 = \pm 3
$$

Solve for $$ x $$:

$$
x = 6 \pm 3
$$

So, $$ x = 9 $$ or $$ x = 3 $$.

**Answer:** The missing $$ x $$-coordinate is either **3** or **9**.

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### **3. Calculate the midpoint of line segment $$ EF $$ where $$ E(-4, 7) $$ and $$ F(2, 5) $$.**

**Solution:**

The **midpoint formula** finds the point exactly in the middle of two points $$ E(x_1, y_1) $$ and $$ F(x_2, y_2) $$:

$$
\text{Midpoint} \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
$$

**Applying the formula:**

$$
x_1 = -4,\ y_1 = 7 \
x_2 = 2,\ y_2 = 5
$$

$$
\text{Midpoint} = \left( \frac{-4 + 2}{2}, \frac{7 + 5}{2} \right) \
= \left( \frac{-2}{2}, \frac{12}{2} \right) \
= (-1, 6)
$$

**Answer:** The midpoint of $$ EF $$ is **$$(-1, 6)$$**.

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I hope this clears up your problems! Let me know if you have any further questions.
1. The length of $$ CD $$ is $$ 2\sqrt{13} $$ units, approximately **7.21 units**. 2. The missing $$ x $$-coordinate is either **3** or **9**. 3. The midpoint of $$ EF $$ is **$$(-1, 6)$$**.

1. Calculate the length of \( C D \) if \( C(-9 ; 1) \) and \( D(-3 ; 5) \). 2f the length of \( A B \) is 5 , calculate the missing coordinate if \( A(6 ;-2) \) and \( B(x ; 2) \). 3. Calculate the midpoint of line segment \( E F \) if \( E(-4 ; 7) \) and \( F(2 ; 5) \).

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