The legs of a right-angled triangle are 6: 8, and its hypotenuse is 1 dm. Find the perimeter of the triangle.

Let the length of the first leg be 6x dm, and the second 8x dm. Then we can write an expression for finding the hypotenuse according to the Pythagorean theorem:

1 = 6x ^ 2 + 8x ^ 2;

Let’s simplify the expression:

1 = 100x ^ 2.

Let’s take the square root of the left and right sides of the equation:

1 = 10x

From here we find x:

x = 1:10;

x = 0.1 dm.

Hence, the first leg is equal to:

0.1 * 6 = 0.6 dm

And second:

0.1 * 8 = 0.8 dm

Now we can find the perimeter of the triangle – add the lengths of the legs and the hypotenuse:

0.6 + 0.8 + 1 = 2.4 dm.

Answer: The perimeter of the triangle is 2.4 dm.