To find the perimeter of the dilated triangle, we first need to calculate the lengths of the sides of the original triangle.
The sides given are:
- One leg: 24 cm
- Another leg: 7 cm
To find the hypotenuse \( c \) of the right triangle, we can use the Pythagorean theorem:
\[ c = \sqrt{a^2 + b^2} \]
where \( a = 24 \) cm and \( b = 7 \) cm.
Calculating the hypotenuse:
\[ c = \sqrt{24^2 + 7^2} = \sqrt{576 + 49} = \sqrt{625} = 25 \text{ cm} \]
Now we have the original triangle sides:
- Leg 1 = 24 cm
- Leg 2 = 7 cm
- Hypotenuse = 25 cm
Next, we find the perimeter of the original triangle by summing the lengths of its sides:
\[ \text{Perimeter} = 24 + 7 + 25 = 56 \text{ cm} \]
Now that we have the perimeter of the original triangle, we apply the dilation with a scale factor of \( \frac{1}{4} \).
To find the perimeter of the dilated triangle, we multiply the original perimeter by the scale factor:
\[ \text{Perimeter of dilated triangle} = \frac{1}{4} \times 56 = 14 \text{ cm} \]
Therefore, the perimeter of the dilated triangle is \( \boxed{14} \text{ cm} \).
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