Given an isosceles triangle where the altitude divides the triangle into two congruent right triangles, we can use the properties of right triangles to find the lengths of the sides and subsequently the perimeter.
Let's denote the following:
- The length of the altitude (height) from the vertex to the base is \( h = 5 \) inches.
- The length of the base of the triangle (the segment cut by the altitude) is \( b = 2 \) inches.
Since the altitude cuts the base into two equal segments, each segment of the base is: \[ \frac{b}{2} = \frac{2}{2} = 1 \text{ inch}. \]
Now we have a right triangle formed by the altitude, half the base, and the congruent side of the isosceles triangle. We can label the lengths:
- One segment of the base (half of it) = \( 1 \) inch.
- The altitude = \( 5 \) inches.
- The side of the triangle that we need to find = \( s \).
Using the Pythagorean theorem for one of the right triangles, we can write: \[ s^2 = h^2 + \left(\frac{b}{2}\right)^2. \]
Substituting the known values: \[ s^2 = 5^2 + 1^2, \] \[ s^2 = 25 + 1, \] \[ s^2 = 26. \]
Taking the square root to find \( s \): \[ s = \sqrt{26}. \]
To find the perimeter \( P \) of the isosceles triangle, we sum the lengths of all three sides: \[ P = 2s + b = 2\sqrt{26} + 2. \]
Calculating \( 2\sqrt{26} \): \[ \sqrt{26} \approx 5.099, \] \[ 2\sqrt{26} \approx 2 \times 5.099 \approx 10.198. \]
Now, adding \( 2 \): \[ P \approx 10.198 + 2 \approx 12.198. \]
Rounding to the nearest tenth, the perimeter is: \[ \boxed{12.2} \text{ inches}. \]
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