To find the value of \(x\) in the right triangle \(MNP\) with altitude \(MO\), we need to apply the properties of right triangles and the formula for the area.
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Identify the Lengths: Let's assume that the lengths are:
- \(MN = 7\)
- \(NP = 5\)
We need to determine \(MP\), the length of the hypotenuse of triangle \(MNP\). We can calculate it using the Pythagorean theorem:
\[ MP = \sqrt{MN^2 + NP^2} = \sqrt{7^2 + 5^2} = \sqrt{49 + 25} = \sqrt{74} \]
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Area of Triangle \(MNP\): The area \(A\) of triangle \(MNP\) can be calculated using the formula for the area of a right triangle:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times MN \times NP \]
Substituting the values:
\[ A = \frac{1}{2} \times 7 \times 5 = \frac{35}{2} \]
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Finding the Length of MO: The area can also be expressed in terms of the hypotenuse \(MP\) and the altitude \(MO\) drawn to the hypotenuse:
\[ A = \frac{1}{2} \times MP \times MO \]
We need to find \(MO\):
\[ MO = \frac{2A}{MP} \]
Substituting \(A\) and \(MP\):
\[ MO = \frac{2 \times \frac{35}{2}}{\sqrt{74}} = \frac{35}{\sqrt{74}} \]
To express this in improper fraction form, multiply the numerator and denominator by \(\sqrt{74}\):
\[ MO = \frac{35\sqrt{74}}{74} \]
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Conclusion: The value of \(x\), which is the length of \(MO\), is given by:
\[ x = \frac{35\sqrt{74}}{74} \]
This represents the exact answer in improper fraction form.
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