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 the hypotenuse

In a right triangle, by the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(c\) is the hypotenuse. If \(x\) is the hypotenuse, then \(a = 4\), \(b=5\). So \(x^{2}=4^{2}+5^{2}=16 + 25=41\), then \(x=\sqrt{41}\approx6.4\) (rounded to the nearest tenth).

Step2: Case 2 - x is a leg

If \(x\) is a leg, then the hypotenuse is \(5\) (since \(5>4\)). So by Pythagorean theorem, \(x^{2}+4^{2}=5^{2}\), \(x^{2}=25 - 16 = 9\), then \(x = 3.0\) (since length is positive).

Step3: Find the difference

The two possible values of \(x\) are approximately \(6.4\) and \(3.0\). The difference is \(6.4-3.0 = 3.4\).

Answer:

3.4 units