7

5
N
O
M
P
Given is the right triangle MNP
with altitude MO
. Find the value of x. Provide an exact answer in improper fraction form.

(1 point)
x=

1 answer

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.

  1. 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:

    \[ MP = \sqrt{MN^2 + NP^2} = \sqrt{7^2 + 5^2} = \sqrt{49 + 25} = \sqrt{74} \]

  2. 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:

    \[ A = \frac{1}{2} \times 7 \times 5 = \frac{35}{2} \]

  3. 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 \(\sqrt{74}\):

    \[ MO = \frac{35\sqrt{74}}{74} \]

  4. 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.

Similar Questions
  1. 9. Find the length of the altitude drawn to the hypotenuse.Triangle 1 is a right triangle with an altitude of 4 Triangle 2 is a
    1. answers icon 1 answer
  2. 9. Find the length of the altitude drawn to the hypotenuse.Triangle 1 is a right triangle with an altitude of 4 Triangle 2 is a
    1. answers icon 1 answer
  3. An Altitude ...• Originates 1. of the triangle at a 2. • It connects the vertex to the line containing the opposite side at
    1. answers icon 1 answer
  4. An Altitude ...• Originates 1. ____ of the triangle at a 2. _____ • It connects the vertex to the line containing the
    1. answers icon 1 answer
more similar questions