draw a neat diagram for each question. 

a) martin walked from home (H) to the local store (S) on a bearing of 035 degrees. he then wlaked on a bearing of 125 degrees to the park ( P) which is 450m due east of his home.

i) show that angle HSP = 90 degrees

ii ) find the distance between martin's home and the store (1dp)

To solve the problem, let's break it down into parts and draw a neat diagram that represents the scenario.

### **Diagram Explanation:**
1. **Point H**: Represents Martin's home.
2. **Point S**: Represents the local store.
3. **Point P**: Represents the park.
4. **\( \overrightarrow{HS} \)**: The path from home to the store with a bearing of 035°.
5. **\( \overrightarrow{SP} \)**: The path from the store to the park with a bearing of 125°.
6. **\( \overrightarrow{HP} \)**: The line directly from home to the park, which is due east and is given as 450m.

Let's proceed to solve the problem and draw the diagram. 

### **Step-by-Step Solution:**

**Part (i): Show that angle HSP = 90 degrees**

- **Bearing** is measured clockwise from the North. So:
  - Bearing from H to S is 035°.
  - Bearing from S to P is 125°.

  Since \( \angle HSP \) represents the change in direction between the two paths:

  \[
  \angle HSP = 125° - 35° = 90°
  \]

Thus, \( \angle HSP \) is a right angle, meaning angle HSP = 90°.

**Part (ii): Find the distance between Martin's home and the store (HS)**

Now we know the triangle \( \triangle HSP \) is a right triangle with:
- \( HP = 450 \) m (due east).
- \( \angle HSP = 90° \).

Using trigonometry (specifically, the sine rule):

\[
HS = HP \times \tan(\text{angle SHP})
\]
Since the angle SHP is 35°:
\[
HS = 450 \times \tan(35°)
\]

Now, let's compute the value.

---

I'll provide the diagram and then solve for HS.### **Solution:**

The distance between Martin's home and the store (HS) is approximately **315.1 meters**.

---

### **Diagram Description:**

To represent this scenario:
1. Draw point **H** for Martin's home.
2. From **H**, draw a line towards **S** making an angle of 35° with the north (line pointing upwards).
3. Then from **S**, draw a line towards **P** making an angle of 90° (since angle HSP is 90°), and place **P** 450 meters directly east of **H**.
4. Label the points H, S, and P accordingly.

This right triangle diagram will illustrate the given problem, where \( \angle HSP = 90^\circ \).

---

Would you like to see the diagram or have any further questions? Here are some related questions you might explore:

1. How can you use the sine or cosine rules in right-angled triangles?
2. What is the general method to solve a problem using bearings?
3. How do you find the shortest distance between two points given their coordinates?
4. How do you calculate the area of a triangle using trigonometric functions?
5. How does changing the bearing angle affect the distance in navigation problems?

**Tip:** When dealing with navigation problems, always visualize the path by drawing a diagram. This helps in understanding the relationship between angles and distances.

Learn how to solve a navigation problem involving bearings and right triangles. Calculate the angle HSP, demonstrate it's 90 degrees, and find the distance between Martin's home and the local store using trigonometric functions. Suitable for high school students.

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