The perimeter of the original right triangle is 40 units. This is determined by halving the perimeter of the dilated triangle, as each side length in the original triangle is half the corresponding length in the dilated triangle.
To find the perimeter of the original right triangle, we can use the scale factor and the perimeter of the dilated triangle.
Step 1: Calculate the perimeter of the dilated triangle.
The perimeter is the sum of the side lengths: 30 + 34 + 16 = 80 units.
Step 2: Apply the scale factor to find the side lengths of the original triangle.
Each side length of the original triangle is half the length of the corresponding side in the dilated triangle. Therefore, the sides of the original triangle are:
Short leg: 30 units / 2 = 15 units
Long leg: 34 units / 2 = 17 units
Hypotenuse: 16 units / 2 = 8 units
Step 3: Calculate the perimeter of the original triangle.
Add the side lengths of the original triangle: 15 units + 17 units + 8 units = 40 units.
Therefore, the perimeter of the preimage, the original right triangle, is 40 units.
Alternatively:
We can use the fact that the perimeter of a dilated figure is equal to the perimeter of the original figure multiplied by the scale factor. In this case, the perimeter of the original triangle is 1/2 the perimeter of the dilated triangle:
Perimeter of original triangle = Perimeter of dilated triangle / Scale factor
Perimeter of original triangle = 80 units / 2
Perimeter of original triangle = 40 units
Therefore, both methods lead to the same answer: the perimeter of the original right triangle is 40 units.