Let's consider the two connected right triangles. In each triangle, we'll denote the short leg as \( a = 4 \) and the long leg as \( b = 16 \).
In a right triangle, the height (altitude) from the right angle to the hypotenuse can be calculated using the formula for the area of the triangle. The area can be calculated in two ways: using the legs and using the hypotenuse and the height.
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Area using the legs: \[ \text{Area} = \frac{1}{2} \times a \times b = \frac{1}{2} \times 4 \times 16 = 32 \]
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Area using the hypotenuse and altitude: The hypotenuse \( c \) can be calculated using the Pythagorean theorem: \[ c = \sqrt{a^2 + b^2} = \sqrt{4^2 + 16^2} = \sqrt{16 + 256} = \sqrt{272} = 4\sqrt{17} \]
The area can also be expressed as: \[ \text{Area} = \frac{1}{2} \times c \times x = \frac{1}{2} \times (4\sqrt{17}) \times x \]
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Equating the two expressions for area: \[ 32 = \frac{1}{2} \times (4\sqrt{17}) \times x \] Multiplying both sides by 2: \[ 64 = 4\sqrt{17} \times x \] Solving for \( x \): \[ x = \frac{64}{4\sqrt{17}} = \frac{16}{\sqrt{17}} \]
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Rationalizing the denominator: \[ x = \frac{16\sqrt{17}}{17} \]
So, the altitude \( x \) is: \[ \boxed{\frac{16\sqrt{17}}{17}} \]
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