LO−→
bisects <NLM, LM = 12, NO = 3 and LN = 8. What is the value of x? Drawing is not to scale. Please show ALL work in order to receive full credit for the problem.
To find the value of x, we can use the angle bisector theorem.
According to the angle bisector theorem, the ratio of the lengths of the two segments formed by an angle bisector is equal to the ratio of the lengths of the two sides opposite the angle.
In this case, we have NO = 3 and LN = 8. Let's call the length of x as LO.
Using the angle bisector theorem, we can set up the following equation:
LO/LN = NO/NM
Substituting the given values, we have:
LO/8 = 3/NM
To find NM, we can use the Pythagorean theorem.
From the given information, we have LM = 12, LO = x, and NO = 3. By applying the Pythagorean theorem to the right triangle NLM, we get:
NM^2 = NL^2 + LM^2
NM^2 = 8^2 + 12^2
NM^2 = 64 + 144
NM^2 = 208
NM = sqrt(208)
NM ≈ 14.42
Now we can substitute the values into the equation:
x/8 = 3/14.42
To solve for x, we can cross-multiply:
14.42x = 8 * 3
14.42x = 24
Dividing both sides by 14.42:
x = 24/14.42
x ≈ 1.66
Therefore, the value of x is approximately 1.66.