We start with the isosceles triangle, which has an altitude that divides it into two congruent right triangles. The base of the triangle is 17 inches, which is divided into two equal segments by the altitude. Therefore, each segment of the base measures:
\[ \frac{17}{2} = 8.5 \text{ inches} \]
The altitude from the vertex to the base measures 18 inches. In each of the right triangles formed by the altitude, we have:
- One leg is the altitude, which is 18 inches.
- The other leg is half of the base, which is 8.5 inches.
To find the lengths of the equal sides of the isosceles triangle (the hypotenuse of the right triangles), we can use the Pythagorean theorem. Let \( s \) represent the length of each of the equal sides. According to the Pythagorean theorem:
\[ s^2 = (8.5)^2 + (18)^2 \]
Calculating \( (8.5)^2 \):
\[ (8.5)^2 = 72.25 \]
Calculating \( (18)^2 \):
\[ (18)^2 = 324 \]
Now summing these values:
\[ s^2 = 72.25 + 324 = 396.25 \]
Taking the square root to find \( s \):
\[ s = \sqrt{396.25} \approx 19.93 \text{ inches} \]
Since there are two equal sides in the isosceles triangle, we find the total perimeter \( P \) of the triangle:
\[ P = 2s + \text{base} = 2(19.93) + 17 \]
Calculating \( 2(19.93) \):
\[ 2(19.93) \approx 39.86 \]
So the perimeter \( P \) becomes:
\[ P \approx 39.86 + 17 = 56.86 \text{ inches} \]
Rounding to the nearest tenth, the perimeter of the triangle is:
\[ \boxed{56.9} \text{ inches} \]
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