[1 mak) A line segrnent joins the points $$ C(-5.6) $$ and $$ D(7,2) $$. (a) Colculate the midpoint of the Ine segment CD $$ \begin{array}{l} 17\left(\frac{-1+1+1}{2}+\frac{1+12}{2}\right) \ m_{2}=-5+7=\frac{2}{2}= \ N_{y}=\frac{\operatorname{Len} 2}{2}=-7= \ m=(-1,4) \end{array} $$ $$ \qquad $$ (2 man (b) Find the gradient of the line segment $$ C D $$. $$ \qquad $$ (2 ms (c) Another line, $$ A B $$, is parallel to $$ C D $$ and passes through the point $$ (0,1) $$. Write down the equation of the line $$ A B $$.

Question

UpStudy AI · Accepted Answer

Let's solve each part of the problem step by step.

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**Problem 1**

*Points:*
- $$ C(-5, 6) $$
- $$ D(7, 2) $$

### (a) Calculate the midpoint of the line segment $$ CD $$.

**Solution:**

The midpoint $$ M $$ of a line segment connecting two points $$ C(x_1, y_1) $$ and $$ D(x_2, y_2) $$ is given by the formula:

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

**Applying the formula:**

$$
M_x = \frac{-5 + 7}{2} = \frac{2}{2} = 1
$$
$$
M_y = \frac{6 + 2}{2} = \frac{8}{2} = 4
$$

**Midpoint:**

$$
M = (1, 4)
$$

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### (b) Find the gradient of the line segment $$ CD $$.

**Solution:**

The gradient (slope) $$ m $$ of a line passing through points $$ C(x_1, y_1) $$ and $$ D(x_2, y_2) $$ is calculated as:

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

**Applying the formula:**

$$
m = \frac{2 - 6}{7 - (-5)} = \frac{-4}{12} = -\frac{1}{3}
$$

**Gradient:**

$$
m = -\frac{1}{3}
$$

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### (c) Write down the equation of the line $$ AB $$ which is parallel to $$ CD $$ and passes through the point $$ (0, 1) $$.

**Solution:**

Since line $$ AB $$ is parallel to line $$ CD $$, it will have the same gradient. From part (b), the gradient $$ m = -\frac{1}{3} $$.

Using the **point-slope form** of the equation of a line:

$$
y - y_1 = m(x - x_1)
$$

Given point $$ (0, 1) $$:

$$
y - 1 = -\frac{1}{3}(x - 0)
$$
$$
y - 1 = -\frac{1}{3}x
$$
$$
y = -\frac{1}{3}x + 1
$$

**Equation of line $$ AB $$:**

$$
y = -\frac{1}{3}x + 1
$$

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**Summary of Answers:**

(a) **Midpoint of $$ CD $$:** $$ M = (1, 4) $$

(b) **Gradient of $$ CD $$:** $$ m = -\frac{1}{3} $$

(c) **Equation of line $$ AB $$:** $$ y = -\frac{1}{3}x + 1 $$
Midpoint of CD is (1, 4). The gradient of CD is -1/3. The equation of line AB is y = -1/3x + 1.

[1 mak)

A line segrnent joins the points and .
(a) Colculate the midpoint of the Ine segment CD

(2 man
(b) Find the gradient of the line segment .

(2 ms
© Another line, , is parallel to and passes through the point . Write down the equation of the line .

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[1 mak) A line segrnent joins the points and . (a) Colculate the midpoint of the Ine segment CD (2 man (b) Find the gradient of the line segment . (2 ms © Another line, , is parallel to and passes through the point . Write down the equation of the line .

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Answer

Solution

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[1 mak)

A line segrnent joins the points and .
(a) Colculate the midpoint of the Ine segment CD

(2 man
(b) Find the gradient of the line segment .

(2 ms
© Another line, , is parallel to and passes through the point . Write down the equation of the line .