https://file.aiquickdraw.com/mathbot/file/1729414387664560bq1tf.jpg

Let's analyze the problem described in the image.

We are dealing with a triangle called a "50–60–70 triangle," where two of the side lengths are different numbers taken from the set \(\{50, 60, 70\}\), and the remaining number represents one of the angles in degrees.

To find the triangle with the smallest perimeter, we need to consider all possible combinations of two side lengths and calculate the perimeter for each case to determine which one is the smallest.

### Possible Combinations:
The possible side combinations with their corresponding angle are:
1. Sides: 50 mm, 60 mm; Angle: 70 degrees
2. Sides: 50 mm, 70 mm; Angle: 60 degrees
3. Sides: 60 mm, 70 mm; Angle: 50 degrees

For each of these cases, we can use the Law of Cosines to calculate the third side \(c\) and then find the perimeter of the triangle.

The Law of Cosines formula is:
\[
c^2 = a^2 + b^2 - 2ab \cdot \cos(C)
\]
where:
- \(a\) and \(b\) are the given side lengths
- \(C\) is the angle between them
- \(c\) is the third side

Let's calculate the perimeter for each case using this formula.

I'll proceed with the calculations.The perimeters of the three possible triangles are as follows:

1. Sides 50 mm, 60 mm; Angle 70 degrees: Perimeter = 173.6229 mm
2. Sides 50 mm, 70 mm; Angle 60 degrees: Perimeter = 182.45 mm
3. Sides 60 mm, 70 mm; Angle 50 degrees: Perimeter = 185.6829 mm

The triangle with the smallest perimeter is the one with sides 50 mm and 60 mm, and an angle of 70 degrees, with a perimeter of \( P = 173.6 \) mm (to 4 significant figures).

Would you like more details or have any questions?

### Related Questions:
1. How do we apply the Law of Cosines to find the missing side in a triangle?
2. What are the conditions for a triangle to exist based on its sides and angles?
3. How does changing an angle in a triangle affect its perimeter?
4. What is the significance of using significant figures in mathematical problems?
5. Are there other methods to determine the smallest perimeter without calculations?

### Tip:
Always check the triangle inequality when dealing with possible triangle side lengths to ensure they can actually form a valid triangle.

A 50-60-70 triangle is a triangle in which two of the side lengths, measured in millimetres, are different numbers taken from the set {50, 60, 70}, with the remaining number being one of the angles in the triangle, measured in degrees. Of all 50-60-70 triangles, one has a smaller perimeter than the rest. If this perimeter is P mm, what is P (to 4 significant figures)?

Discover how to find the smallest perimeter of a 50-60-70 triangle using the Law of Cosines. This detailed step-by-step solution is suitable for high school students (Grades 9-12) and explores trigonometry, including side length calculations and perimeter estimation.

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