https://file.aiquickdraw.com/mathbot/file/1728405429114pndl2l3a.jpg

Let's break down and solve each part of the question step by step.

### Part (a)
Emily wants to put a walkway from **Point A** to **Point G**. The problem asks for the length of the path between these points, and we are instructed to use the **distance formula**. The distance formula is given by:

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Where:
- \((x_1, y_1)\) are the coordinates of Point A
- \((x_2, y_2)\) are the coordinates of Point G

From the grid:
- Point A has coordinates \((-6, 4)\)
- Point G has coordinates \((2, -3)\)

Now, substitute these values into the distance formula:

\[
d = \sqrt{(2 - (-6))^2 + (-3 - 4)^2} = \sqrt{(2 + 6)^2 + (-3 - 4)^2}
\]
\[
d = \sqrt{8^2 + (-7)^2} = \sqrt{64 + 49} = \sqrt{113}
\]

Thus, the distance is:

\[
d = \sqrt{113} \approx 10.63 \text{ units}
\]

Each unit represents **one yard**, so the length of the walkway is approximately **10.63 yards**.

---

### Part (b)
Emily wants to create a second walkway starting at Point A that is **perpendicular** to the walkway connecting Point A to Point G. To find the slope of this perpendicular walkway, we first need the slope of the line connecting **Point A** to **Point G**.

The slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Substitute the coordinates of Points A \((-6, 4)\) and G \((2, -3)\):

\[
m = \frac{-3 - 4}{2 - (-6)} = \frac{-7}{2 + 6} = \frac{-7}{8}
\]

Now, the slope of the second walkway, which is perpendicular to this line, is the **negative reciprocal** of this slope. So, if the slope of the line connecting A and G is \(\frac{-7}{8}\), the perpendicular slope is:

\[
m_{\text{perpendicular}} = \frac{8}{7}
\]

---

### Part (c)
We need to find the equation of the line for the second walkway, which has a slope of \(\frac{8}{7}\) and passes through Point A \((-6, 4)\).

The equation of a line can be written in **point-slope form**:

\[
y - y_1 = m(x - x_1)
\]

Where \((x_1, y_1)\) is Point A \((-6, 4)\) and \(m = \frac{8}{7}\). Substituting these values:

\[
y - 4 = \frac{8}{7}(x - (-6)) = \frac{8}{7}(x + 6)
\]

Now, expand the equation:

\[
y - 4 = \frac{8}{7}x + \frac{48}{7}
\]

Add 4 to both sides (4 is equivalent to \(\frac{28}{7}\)):

\[
y = \frac{8}{7}x + \frac{48}{7} + \frac{28}{7}
\]
\[
y = \frac{8}{7}x + \frac{76}{7}
\]

Thus, the equation of the second walkway is:

\[
y = \frac{8}{7}x + \frac{76}{7}
\]

---

### Recap:
- **Part (a)**: The walkway from A to G is approximately **10.63 yards**.
- **Part (b)**: The slope of the second, perpendicular walkway is **\(\frac{8}{7}\)**.
- **Part (c)**: The equation of the second walkway is \(y = \frac{8}{7}x + \frac{76}{7}\).

Let me know if you'd like further clarification or more details on any of the steps!

---

**Relative questions:**
1. How would the distance formula change if the walkway were in three-dimensional space?
2. What would happen if the slopes of two lines were not perpendicular but parallel?
3. How can the equation of the line be rewritten in standard form?
4. What is the geometrical significance of the distance between points on a graph?
5. How can the perpendicular bisector of the segment AG be found?

**Tip**: Always double-check that you're using the correct points when applying formulas to avoid errors.

Emily is a city planner tasked with creating a new city park that includes a garden and a playground. The problem asks her to calculate the distance between two points on a coordinate grid using the distance formula and determine the slope and equation of a line perpendicular to the given walkway between two points.

Learn how to solve a city park planning problem involving the distance formula, slope, and equation of a line. In this problem, you will calculate the distance between two points on a coordinate grid and find the equation of a perpendicular line. Suitable for students in Grades 9-10.

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