Points A, B and C represent position of three towers such that AB = 60 m, BC = 73 m and CA 52 m. Taking a scale of 10 m to 1 cm, make an accurate drawing of AABC. Find by drawing, the location of a point which is equidistant from A, B and C, and its actual distance from any of the towers.

To solve this problem, we need to accurately draw the triangle \( \triangle ABC \) based on the given distances and then find the circumcenter of the triangle, which is the point equidistant from all three vertices (A, B, and C).

### Step-by-Step Solution

#### 1. **Choose a Scale and Convert Distances**
Given scale: **10 m to 1 cm**

- \(AB = 60 \, \text{m} \implies 6 \, \text{cm} \)
- \(BC = 73 \, \text{m} \implies 7.3 \, \text{cm} \)
- \(CA = 52 \, \text{m} \implies 5.2 \, \text{cm} \)

#### 2. **Draw Triangle \( \triangle ABC \)**

1. **Draw Line Segment \(AB\):**
   - Draw a line segment \(AB = 6 \, \text{cm}\) using a ruler.

2. **Locate Point C using Compass:**
   - Set your compass to a radius of \(5.2 \, \text{cm}\) (for \(CA\)). Place the compass point on \(A\) and draw an arc.
   - Next, set the compass to a radius of \(7.3 \, \text{cm}\) (for \(BC\)). Place the compass point on \(B\) and draw another arc, intersecting the first arc.
   - Mark the intersection point as \(C\).

3. **Connect Points:**
   - Draw lines \(AC\) and \(BC\) to complete the triangle \( \triangle ABC \).

#### 3. **Locate the Circumcenter**

The circumcenter is the point where the perpendicular bisectors of the sides of the triangle intersect.

1. **Draw the Perpendicular Bisector of \(AB\):**
   - Find the midpoint of \(AB\) (which is \(3 \, \text{cm}\) from both A and B).
   - Draw a perpendicular line from this midpoint using a right-angle ruler or compass.

2. **Draw the Perpendicular Bisector of \(BC\):**
   - Find the midpoint of \(BC\) (which is \(3.65 \, \text{cm}\) from both B and C).
   - Draw a perpendicular line from this midpoint.

3. **Intersection Point:**
   - The intersection of these two perpendicular bisectors is the **circumcenter** \(O\) of \( \triangle ABC \).

#### 4. **Measure the Distance from the Circumcenter to Any Vertex**

Using the drawing, measure the distance from \(O\) to any of the three points (A, B, or C) with a ruler. Let's say it measures \(d\) cm on the drawing.

#### 5. **Convert Back to Actual Distance**

Convert this measurement back to meters using the scale (10 m to 1 cm).

\[
\text{Actual distance} = d \times 10 \, \text{m}
\]

### Conclusion

After completing the steps:

1. The drawing provides a visual representation of \( \triangle ABC \).
2. The circumcenter \(O\) is identified as the point equidistant from A, B, and C.
3. The actual distance from \(O\) to any of the towers (A, B, or C) is calculated by measuring on the drawing and converting using the scale.

If you have any questions or need further clarifications on any step, please let me know!

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**Related Questions:**

1. What is the significance of the circumcenter in a triangle?
2. How can the circumcenter be used in real-world applications, such as in construction or navigation?
3. What are the properties of the perpendicular bisectors in a triangle?
4. How do different types of triangles (acute, right, obtuse) affect the location of the circumcenter?
5. How can you construct the incenter, centroid, or orthocenter of a triangle?

**Tip:** Remember that the circumcenter can lie inside, outside, or on the triangle, depending on the type of triangle (acute, obtuse, or right).

Learn how to draw triangle ABC accurately using a scale of 10 m to 1 cm. Find the circumcenter, which is equidistant from points A, B, and C, and determine its actual distance from any tower using the scale conversion. Suitable for high school students studying triangle geometry.

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