Question

consider the work shown to determine the value of x in the following triangle.
triangle image with 30° angle, right angle, base 8√10, hypotenuse x
since this right triangle has a 30° angle, it is a 30°–60°–90° special triangle.
the hypotenuse is √3 times larger than the longer leg.

x = 8√10 √3
x = 8√30

which statement best describes the work shown?
○ this is not a 30°–60°–90° special triangle.
○ the hypotenuse is not √3 times longer than the longer leg.
○ the value for x was not simplified properly.
○ there was no error made.

correctly solve for x. if there was no error, enter the value for x from the work above.

Explanation:

1. Analyze the 30 - 60 - 90 triangle ratios: In a 30° - 60° - 90° triangle, the sides are in the ratio \(1:\sqrt{3}:2\) (shorter leg : longer leg : hypotenuse). So the hypotenuse is \( \frac{2}{\sqrt{3}}=\frac{2\sqrt{3}}{3}\) times the longer leg, not \( \sqrt{3}\) times. The given work incorrectly states the hypotenuse - longer leg relationship.
2. For the first part (multiple - choice): The correct statement is "The hypotenuse is not \( \sqrt{3}\) times longer than the longer leg."
3. For solving \(x\) (hypotenuse): In a 30° - 60° - 90° triangle, \(\cos(30^{\circ})=\frac{\text{adjacent}}{\text{hypotenuse}}\). The adjacent side to the 30° angle is \(8\sqrt{10}\), and \(\cos(30^{\circ})=\frac{\sqrt{3}}{2}\). So \(\frac{\sqrt{3}}{2}=\frac{8\sqrt{10}}{x}\), cross - multiply to get \(x\times\sqrt{3}=2\times8\sqrt{10}\), then \(x = \frac{16\sqrt{10}}{\sqrt{3}}=\frac{16\sqrt{30}}{3}\) (rationalizing the denominator).

Answer:

1. The correct option for the multiple - choice: The hypotenuse is not \(\boldsymbol{\sqrt{3}}\) times longer than the longer leg.
2. The correct value of \(x\) is \(\boldsymbol{\frac{16\sqrt{30}}{3}}\)