Question

a right triangle has side lengths 4 units, 5 units, and x units. it is unknown if the missing length is the longest or shortest side. rounded to the nearest tenth, what is the difference between the possible values of x?
○ 3.0 units
○ 3.4 units
○ 6.4 units
○ 8.0 units

Explanation:

Step1: Case 1 - x is hypotenuse

In a right triangle, by Pythagorean theorem \(a^2 + b^2 = c^2\) (where \(c\) is hypotenuse). Here, \(a = 4\), \(b = 5\)? Wait, no: if x is the hypotenuse, then the other two sides are 4 and 5. So \(x=\sqrt{4^2 + 5^2}=\sqrt{16 + 25}=\sqrt{41}\approx6.4\) (rounded to nearest tenth).

Step2: Case 2 - x is a leg (5 is hypotenuse)

Then, by Pythagorean theorem, \(x=\sqrt{5^2 - 4^2}=\sqrt{25 - 16}=\sqrt{9}=3.0\).

Step3: Find the difference

Subtract the two possible values: \(6.4 - 3.0 = 3.4\).

Answer:

3.4 units