You should use trigonometry, not scale drawings, to find your answers.
In an archaeological expedition, Stephanie is studying the layout of an
ancient city. She has found three points of interest and straight roads joining
these, forming a triangle. One road runs east-west and has a temple at one
end, at point T, and a marketplace at point M, 300 metres due west of the
temple. Standing at the marketplace, looking westwards (with her back to
the temple), Stephanie can see a castle, C, towards the northwest. Stephanie
has to turn 60◦
to face the castle. There is a road MC joining the
marketplace and the castle which is 450 metres long. A third road, CT, runs
from the castle to the temple. You may assume that the terrain is flat and
all roads are along straight lines.
(a) Draw a sketch, including the triangle CMT, marking the angle and the
lengths (in metres) that you are given. [2]
(b) (i) Find the internal angle ∠CMT. [1]
(ii) Find the distance between the castle and the temple, that is, the
length of the side CT. Give your answer correct to the nearest
metre. [3]
(iii) Stephanie has been looking for other ancient structures and has
found the remains of a dungeon, along the east-west road, between
the marketplace and the temple. The dungeon, D, is 100 metres
due east of the marketplace. There are also signs of a fourth road,
running north from the dungeon and joining the road CT at a
junction J. What is the length of the road DJ? Give your answer
correct to the nearest metre.

### Part (a): Sketch the Triangle CMT

To sketch the triangle, we can mark the given points:

- **T (Temple)** is at the origin of our coordinate system.
- **M (Marketplace)** is 300 meters due west of T.
- **C (Castle)** is located 450 meters from M at an angle of 60° northwest (relative to M).

### Part (b): Trigonometric Calculations

#### (i) Find the internal angle ∠CMT

Since Stephanie turns 60° to face the castle from M, and since M and T are on a straight east-west line, the angle ∠CMT is the same as the angle Stephanie turns.

\[
\text{∠CMT} = 60^\circ
\]

#### (ii) Find the distance between the castle and the temple (CT)

To find the length of CT, we can apply the Law of Cosines in triangle CMT:

\[
CT^2 = CM^2 + MT^2 - 2 \times CM \times MT \times \cos(\text{∠CMT})
\]

Substituting the given values:

\[
CT^2 = 450^2 + 300^2 - 2 \times 450 \times 300 \times \cos(60^\circ)
\]

\[
\cos(60^\circ) = 0.5
\]

So,

\[
CT^2 = 450^2 + 300^2 - 2 \times 450 \times 300 \times 0.5
\]

\[
CT^2 = 202500 + 90000 - 135000 = 157500
\]

\[
CT = \sqrt{157500} \approx 397 \text{ meters}
\]

#### (iii) Find the length of the road DJ

To find the length of the road DJ, we need to use the fact that D is 100 meters east of M along the east-west line. Since DJ is perpendicular to CT and joins at J, we need to calculate the perpendicular distance from D to line CT.

Let's denote:

- \( x \) as the distance from M to the perpendicular point J on CT.

Using trigonometry:

\[
\sin(60^\circ) = \frac{x}{450} \quad \text{and} \quad x = 450 \times \sin(60^\circ)
\]

\[
x = 450 \times \frac{\sqrt{3}}{2} \approx 389.71 \text{ meters}
\]

Now, the length DJ (since D is 100 meters east of M):

\[
DJ = |389.71 - 100| \approx 290 \text{ meters}
\]

### Summary of Results:
- (i) The internal angle ∠CMT is **60°**.
- (ii) The distance between the castle and the temple (CT) is approximately **397 meters**.
- (iii) The length of the road DJ is approximately **290 meters**.

---

Would you like more details or have any further questions? Here are some related questions you might find interesting:

1. How does the Law of Cosines apply in different triangle configurations?
2. What is the significance of the internal angle in solving triangle-related problems?
3. How can trigonometry be used to solve real-world navigation problems?
4. Can the Pythagorean theorem be used in this problem? If so, where?
5. How would the solution change if the roads were not straight lines?
6. What are the practical applications of these trigonometric calculations?
7. How would the scenario change if the terrain was not flat?
8. What other mathematical techniques can be used to solve similar geometry problems?

**Tip:** When using the Law of Cosines, make sure to correctly identify the angles and sides in relation to each other.

Explore a trigonometry problem involving an ancient city layout, where distances and angles are calculated using the Law of Cosines. Learn how to determine internal angles and side lengths in triangles based on given archaeological data. Suitable for Grades 10-12 students and anyone interested in practical trigonometric applications.

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