Let's break this problem into parts, then I'll explain the steps to create the diagrams for each part of the question and solve the problem.

Step 1: Understanding the Problem

• Home (H) is the starting point.
• Store (S) is at a bearing of $035^\circ$ from $H$.
• Park (P) is at a bearing of $125^\circ$ from $S$ and is 450 meters due east of $H$.

Step 2: Diagram Explanation

We'll create two diagrams:

1. Diagram 1 showing the positions of $H$, $S$, and $P$ based on the given bearings.
2. Diagram 2 focusing on the right triangle formed to show $\angle HSP$ and calculate the distance.

Step 3: Solve the Problem

Part (i): Show that $\angle HSP = 90^\circ$

Given:

• The bearing of $S$ from $H$ is $035^\circ$, so the angle $\angle SHP = 35^\circ$ from the north.
• The bearing of $P$ from $S$ is $125^\circ$, which means that from the north at $S$, the direction to $P$ is $125^\circ - 035^\circ = 90^\circ$.

This indicates that $\angle HSP = 90^\circ$, as the angle between the two bearings is a right angle.

Part (ii): Find the Distance Between Martin's Home and the Store

To find the distance $HS$, we can use trigonometry because we have a right triangle:

1. $HP = 450$ meters (since $P$ is due east of $H$).
2. The angle $\angle HSP = 90^\circ$.

Using the sine rule:

$\sin(35^\circ) = \frac{HS}{450}$

Solving for $HS$:

$HS = 450 \times \sin(35^\circ) \approx 450 \times 0.5736 = 258.1 \, \text{meters (to 1 decimal place)}$

Diagrams (to be visualized):

1. Diagram 1:

   • $H$ at the origin.
   • $S$ at a $035^\circ$ angle from $H$ (north is $0^\circ$).
   • $P$ at a $125^\circ$ angle from $S$.
   • $P$ is directly east of $H$ (450m).

2. Diagram 2:

   • Triangle $HSP$ showing the right angle at $S$.
   • Label the sides with $HS = 258.1m$ and $HP = 450m$.

These diagrams will demonstrate the relationship between the points and confirm that $\angle HSP = 90^\circ$ as well as provide the required distance.

Would you like to see a visual representation of these diagrams?