To find the perimeter of the isosceles triangle with altitude, we work through the problem step by step.
Given:
- The length of the altitude (height) \( h = 18 \) inches.
- The length of the base \( b = 17 \) inches.
Since the altitude divides the base into two equal segments, each segment will be: \[ \frac{b}{2} = \frac{17}{2} = 8.5 \text{ inches.} \]
Now, our altitude, base segment, and the side of the triangle can be understood as forming a right triangle. The altitude forms a right triangle with half of the base, where one leg is the height \( h \), the other leg is half of the base \( \frac{b}{2} = 8.5 \) inches, and the hypotenuse is the length of one of the equal sides \( s \) of the isosceles triangle.
Using the Pythagorean theorem, we have: \[ s^2 = h^2 + \left(\frac{b}{2}\right)^2. \] Substituting the known values: \[ s^2 = 18^2 + 8.5^2. \]
Calculating \( 18^2 \) and \( 8.5^2 \): \[ 18^2 = 324, \] \[ 8.5^2 = 72.25. \]
Thus, we can find \( s^2 \): \[ s^2 = 324 + 72.25 = 396.25. \]
Now we take the square root to find \( s \): \[ s = \sqrt{396.25} \approx 19.9 \text{ inches.} \]
Now we can find the perimeter \( P \) of the triangle. The perimeter of the isosceles triangle is given by: \[ P = 2s + b. \] Substituting \( s \) and \( b \) into the formula: \[ P = 2(19.9) + 17, \] \[ P = 39.8 + 17 = 56.8 \text{ inches.} \]
Therefore, the perimeter of the triangle, rounded to the nearest tenth of an inch, is: \[ \boxed{56.8} \text{ inches.} \]
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