Constructing a Triangle with a Perimeter of 160mm and Sides in the Ratio 3:5:6

The problem involves constructing a triangle with a perimeter of 160mm, where the sides are in the ratio 3:5:6. This guide will walk you through the steps to determine the lengths of the sides and construct the triangle using basic tools like a ruler and compass.

Determining the Lengths of the Sides

To start, we need to calculate the length of each side based on the given ratio and perimeter.

First, let's determine the total number of parts in the ratio:

3 5 6 14 parts

Next, we find the length of each part by dividing the perimeter by the total number of parts:

k 16014k??frac{160}{14}

k 16014 80/7k??frac{160}{14}??80/7

Now, we calculate the length of each side:

Side 1 (ratio 3): ( 3k 3 times frac{160}{14} frac{480}{14} frac{240}{7} approx 34.2857 ) mm

Side 2 (ratio 5): ( 5k 5 times frac{160}{14} frac{800}{14} frac{400}{7} approx 57.1429 ) mm

Side 3 (ratio 6): ( 6k 6 times frac{160}{14} frac{960}{14} frac{480}{7} approx 68.5714 ) mm

Verifying the Perimeter

To ensure the perimeter is correct, let's add the calculated lengths of the sides:

34.2857 57.1429 68.5714 160.0000

This confirms that the sum of the sides is indeed 160 mm, with a negligible rounding error.

Constructing the Triangle

Now, let's follow the steps to physically construct the triangle:

Step 1: Draw the First Side

Use a ruler to draw a line segment of approximately 34.2857 mm.

Step 2: Construct the Second Side

At one endpoint of the first side, use a compass to draw an arc with a radius of approximately 57.1429 mm.

Step 3: Construct the Third Side

At the other endpoint of the first side, draw another arc with a radius of approximately 68.5714 mm.

Step 4: Mark the Intersection

The point where the two arcs intersect is the third vertex of the triangle.

Step 5: Connect the Vertices

Use a ruler to connect all three vertices to form the triangle.

Conclusion

You have now successfully constructed a triangle with a perimeter of 160 mm and sides in the ratio 3:5:6.

By following these steps, you can accurately determine and construct a triangle with the specified ratio and perimeter, making this a valuable skill in geometry and construction.

Additional Information

The ratios of the sides can also be expressed as follows:

3n 34.2857 5n 57.1429 6n 68.5714

Where n is the unit scale factor, calculated as:

X 16014 ≈ 11.438571X??frac{160}{14}?≈?11.438571

The three sides are approximately 34.2857 mm, 57.1429 mm, and 68.5714 mm.