2. The points $$ A(2,3), B(1,8), C(6,11) $$ and $$ D(3,6) $$ are the vertices of a quadrilateral. a. Find the gradients of $$ A B $$ and $$ D C $$. What does this tell you about the lines $$ A B $$ and $$ D C $$ ? b. Find the equation of the diagonal joining $$ B $$ to $$ D $$. c. The diagonal $$ A C $$ has the equation $$ y=x+5 $$. Find the point of intersection of the two diagonals $$ A C $$ and $$ B D $$. d. Find the midpoint of $$ A C $$ and compare this point with the point of intersection of the diagonals $$ A C $$ and $$ B D $$.

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UpStudy AI · Accepted Answer

**Problem 2(a):**

*Find the gradients of $$ AB $$ and $$ DC $$. What does this tell you about the lines $$ AB $$ and $$ DC $$?*

**Solution:**

To find the gradients (slopes) of the lines $$ AB $$ and $$ DC $$, we'll use the formula for the slope between two points $$ (x_1, y_1) $$ and $$ (x_2, y_2) $$:

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

1. **Gradient of $$ AB $$:**

- **Points:**
     - $$ A(2, 3) $$
     - $$ B(1, 8) $$

- **Calculation:**
     $$
     m_{AB} = \frac{8 - 3}{1 - 2} = \frac{5}{-1} = -5
     $$
   
2. **Gradient of $$ DC $$:**

- **Points:**
     - $$ D(3, 6) $$
     - $$ C(6, 11) $$

- **Calculation:**
     $$
     m_{DC} = \frac{11 - 6}{6 - 3} = \frac{5}{3}
     $$

**Interpretation:**

- **Comparing Slopes:**
  - $$ m_{AB} = -5 $$
  - $$ m_{DC} = \frac{5}{3} $$

- **Parallel Condition:**
  - Two lines are parallel if their slopes are equal. Here, $$ -5 \neq \frac{5}{3} $$, so $$ AB $$ and $$ DC $$ are **not parallel**.

- **Perpendicular Condition:**
  - Two lines are perpendicular if the product of their slopes is $$ -1 $$. Here:
    $$
    m_{AB} \times m_{DC} = (-5) \times \left(\frac{5}{3}\right) = -\frac{25}{3} \neq -1
    $$
    So, $$ AB $$ and $$ DC $$ are **not perpendicular**.

**Conclusion:**

- The gradient of $$ AB $$ is **-5**.
- The gradient of $$ DC $$ is **$$ \frac{5}{3} $$**.
- Since their slopes are neither equal nor negative reciprocals, the lines $$ AB $$ and $$ DC $$ are **neither parallel nor perpendicular**.

Answer: Problem a Answer

The gradient of $$ AB $$ is $$-5$$ and the gradient of $$ DC $$ is $$ \frac{5}{3} $$. Since these slopes are neither equal nor negative reciprocals, the lines $$ AB $$ and $$ DC $$ are neither parallel nor perpendicular.
**Problem 2(a):** *Find the gradients of $$ AB $$ and $$ DC $$. What does this tell you about the lines $$ AB $$ and $$ DC $$?* **Simplify Answer:** - **Gradient of $$ AB $$:** $$-5$$ - **Gradient of $$ DC $$:** $$\frac{5}{3}$$ Since the gradients are neither equal nor negative reciprocals, lines $$ AB $$ and $$ DC $$ are **neither parallel nor perpendicular**.

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