A line segment joins the points $$ C(-5,6) $$ and $$ \mathbf{D}(7,2) $$. (d) Calculate the midpoint of the line segment $$ \mathbf{C D} $$. (b) Find the gradient of the line segment CD.

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Let's solve each part step by step for the line segment $$ \mathbf{C D} $$ with points $$ C(-5, 6) $$ and $$ D(7, 2) $$.

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### (d) **Midpoint of the Line Segment $$ \mathbf{C D} $$**

The **midpoint** $$ M $$ of a line segment connecting two points $$ (x_1, y_1) $$ and $$ (x_2, y_2) $$ is calculated using the midpoint formula:

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

**Given:**
- Point $$ C(-5, 6) $$: $$ x_1 = -5 $$, $$ y_1 = 6 $$
- Point $$ D(7, 2) $$: $$ x_2 = 7 $$, $$ y_2 = 2 $$

**Calculations:**

$$
\begin{align*}
\text{Midpoint } M_x &= \frac{-5 + 7}{2} = \frac{2}{2} = 1 \
\text{Midpoint } M_y &= \frac{6 + 2}{2} = \frac{8}{2} = 4 \
\end{align*}
$$

**Midpoint $$ M $$:**
$$
M = (1, 4)
$$

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### (b) **Gradient (Slope) of the Line Segment $$ \mathbf{C D} $$**

The **gradient** (or **slope**) $$ m $$ of a line segment between two points $$ (x_1, y_1) $$ and $$ (x_2, y_2) $$ is given by:

$$
m = \frac{y_2 - y_1}{x_2 - x_1}
$$

**Given:**
- Point $$ C(-5, 6) $$: $$ x_1 = -5 $$, $$ y_1 = 6 $$
- Point $$ D(7, 2) $$: $$ x_2 = 7 $$, $$ y_2 = 2 $$

**Calculations:**

$$
\begin{align*}
m &= \frac{2 - 6}{7 - (-5)} \
&= \frac{-4}{12} \
&= -\frac{1}{3}
\end{align*}
$$

**Gradient $$ m $$:**
$$
m = -\frac{1}{3}
$$

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**Summary:**
- **Midpoint of $$ \mathbf{C D} $$:** $$ (1, 4) $$
- **Gradient of $$ \mathbf{C D} $$:** $$ -\frac{1}{3} $$
Midpoint of CD is (1, 4) and the gradient of CD is -1/3.

A line segment joins the points \( C(-5,6) \) and \( \mathbf{D}(7,2) \). (d) Calculate the midpoint of the line segment \( \mathbf{C D} \). (b) Find the gradient of the line segment CD.

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