Question

suppose a triangle with the legs 7 and 24 and the hypotenuse 25 is dilated by a scale factor of 1/4. What is the perimeter of the dilated triangle

Answer 1

To find the perimeter of the dilated triangle, we start with the perimeter of the original triangle.

The original triangle has legs of lengths 7 and 24, and a hypotenuse of 25.

First, we calculate the perimeter of the original triangle:

\[
\text{Perimeter} = \text{leg}_1 + \text{leg}_2 + \text{hypotenuse} = 7 + 24 + 25 = 56
\]

Next, we apply the dilation with a scale factor of \( \frac{1}{4} \). This means that each side of the triangle will be multiplied by \( \frac{1}{4} \).

The new lengths of the sides of the dilated triangle are:

\[
\text{New leg}_1 = 7 \times \frac{1}{4} = \frac{7}{4}
\]
\[
\text{New leg}_2 = 24 \times \frac{1}{4} = 6
\]
\[
\text{New hypotenuse} = 25 \times \frac{1}{4} = \frac{25}{4}
\]

Now we can calculate the perimeter of the dilated triangle:

\[
\text{Dilated Perimeter} = \text{New leg}_1 + \text{New leg}_2 + \text{New hypotenuse}
\]

Substituting the new lengths:

\[
\text{Dilated Perimeter} = \frac{7}{4} + 6 + \frac{25}{4}
\]

To combine these, we first convert 6 into a fraction with a denominator of 4:

\[
6 = \frac{24}{4}
\]

Now we can add all the terms:

\[
\text{Dilated Perimeter} = \frac{7}{4} + \frac{24}{4} + \frac{25}{4} = \frac{7 + 24 + 25}{4} = \frac{56}{4} = 14
\]

Thus, the perimeter of the dilated triangle is:

\[
\boxed{14}
\]