Question

consider the work shown to determine the value of x in the following triangle.
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 = 7√10 × √3
x = 7√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:

Step1: Recall 30-60-90 triangle ratios

In a \(30^\circ - 60^\circ - 90^\circ\) triangle, the sides are in the ratio \(1:\sqrt{3}:2\), where the side opposite \(30^\circ\) is the shortest leg (\(s\)), the side opposite \(60^\circ\) is the longer leg (\(s\sqrt{3}\)), and the hypotenuse is \(2s\). Also, \(\cos(30^\circ)=\frac{\text{adjacent}}{\text{hypotenuse}}\), \(\cos(30^\circ)=\frac{\sqrt{3}}{2}\). Here, the adjacent side to \(30^\circ\) is \(7\sqrt{10}\), and hypotenuse is \(x\). So \(\cos(30^\circ)=\frac{7\sqrt{10}}{x}\), which gives \(x = \frac{7\sqrt{10}}{\cos(30^\circ)}=\frac{7\sqrt{10}}{\frac{\sqrt{3}}{2}}=\frac{14\sqrt{10}}{\sqrt{3}}=\frac{14\sqrt{30}}{3}\). Wait, but first, check the error in the given work. The given work says hypotenuse is \(\sqrt{3}\) times longer than longer leg. But in \(30 - 60 - 90\), hypotenuse is \(\frac{2}{\sqrt{3}}\) times the longer leg (since longer leg is \(s\sqrt{3}\), hypotenuse is \(2s\), so \(2s=\frac{2}{\sqrt{3}}(s\sqrt{3})\)). Or, the adjacent side here is the longer leg (since it's adjacent to \(30^\circ\), so the side opposite \(60^\circ\) is the longer leg). So \(\cos(30^\circ)=\frac{\text{longer leg}}{\text{hypotenuse}}\), so \(\text{hypotenuse}=\frac{\text{longer leg}}{\cos(30^\circ)}=\frac{7\sqrt{10}}{\frac{\sqrt{3}}{2}}=\frac{14\sqrt{10}}{\sqrt{3}}=\frac{14\sqrt{30}}{3}\). The given work incorrectly states that hypotenuse is \(\sqrt{3}\) times longer than longer leg. So the error is in the statement about the ratio.

Step2: Analyze each option

• Option 1: It is a \(30 - 60 - 90\) triangle (right triangle with \(30^\circ\)), so this is wrong.
• Option 2: The hypotenuse is not \(\sqrt{3}\) times longer than longer leg. In \(30 - 60 - 90\), hypotenuse is \(\frac{2}{\sqrt{3}}\) times the longer leg (since longer leg \(= s\sqrt{3}\), hypotenuse \(= 2s\), so \(2s=\frac{2}{\sqrt{3}}(s\sqrt{3})\)). So the given work's statement about hypotenuse being \(\sqrt{3}\) times longer than longer leg is incorrect.
• Option 3: The simplification in given work is \(7\sqrt{10}\times\sqrt{3}=7\sqrt{30}\), which is correct simplification of radicals, but the formula used is wrong. So this option is wrong.
• Option 4: There was an error (wrong ratio), so this is wrong.

Now, to solve for \(x\) correctly. Using \(\cos(30^\circ)=\frac{\text{adjacent}}{\text{hypotenuse}}\), adjacent \(= 7\sqrt{10}\), \(\cos(30^\circ)=\frac{\sqrt{3}}{2}\). So \(x=\frac{7\sqrt{10}}{\frac{\sqrt{3}}{2}}=\frac{14\sqrt{10}}{\sqrt{3}}=\frac{14\sqrt{30}}{3}\) (rationalizing the denominator: multiply numerator and denominator by \(\sqrt{3}\), so \(\frac{14\sqrt{10}\times\sqrt{3}}{3}=\frac{14\sqrt{30}}{3}\)).

(for multiple - choice):

Step1: Recall 30 - 60 - 90 triangle ratios

In a \(30^\circ-60^\circ - 90^\circ\) triangle, the sides are in the ratio \(1:\sqrt{3}:2\) (short leg : long leg : hypotenuse). The long leg (opposite \(60^\circ\)) is \(s\sqrt{3}\), hypotenuse is \(2s\). The cosine of \(30^\circ\) is \(\frac{\text{long leg}}{\text{hypotenuse}}=\frac{\sqrt{3}}{2}\), so \(\text{hypotenuse}=\frac{\text{long leg}}{\frac{\sqrt{3}}{2}}=\frac{2}{\sqrt{3}}\times\text{long leg}\), not \(\sqrt{3}\times\text{long leg}\).

Step2: Analyze each option

• Option 1: It is a \(30 - 60 - 90\) triangle (right triangle with \(30^\circ\)), so eliminate.
• Option 2: As shown, hypotenuse is not \(\sqrt{3}\) times long leg, correct.
• Option 3: \(7\sqrt{10}\times\sqrt{3}=7\sqrt{30}\) is correct simplification, eliminate.
• Option 4: There is an error (wrong ratio), eliminate.

(solving for \(x\)):

Step1: Use cosine ratio

\(\cos(30^\circ)=\frac{\text{adjacent}}{\text{hypotenuse}}\), adjacent \( = 7\sqrt{10}\), hypotenuse \(=x\), \(\cos(30^\circ)=\frac{\sqrt{3}}{2}\). So \(\frac{\sqrt{3}}{2}=\frac{7\sqrt{10}}{x}\).

Step2: Solve for \(x\)

Cross - multiply: \(x\sqrt{3}=14\sqrt{10}\). Then \(x=\frac{14\sqrt{10}}{\sqrt{3}}\). Rationalize: \(x=\frac{14\sqrt{30}}{3}\).

Answer:

The statement "The hypotenuse is not \(\sqrt{3}\) times longer than the longer leg." is correct. So the answer for the first part (which statement) is "The hypotenuse is not \(\sqrt{3}\) times longer than the longer leg." And the correct value of \(x\) is \(\frac{14\sqrt{30}}{3}\).

Wait, the question has two parts? Wait, the original question: "Which statement best describes the work shown?" and "Correctly solve for \(x\)". Let's handle the first part (the multiple choice) first.

For the multiple - choice:

The options are:

1. This is not a \(30^\circ - 60^\circ - 90^\circ\) special triangle. (Incorrect, because it's a right triangle with a \(30^\circ\) angle, so it is a \(30 - 60 - 90\) triangle.)

1. The hypotenuse is not \(\sqrt{3}\) times longer than the longer leg. (Correct, as we saw the ratio is wrong.)

1. The value for \(x\) was not simplified properly. (Incorrect, the simplification of \(7\sqrt{10}\times\sqrt{3}=7\sqrt{30}\) is correct, the error is in the ratio, not simplification.)

1. There was no error made. (Incorrect, since the ratio statement is wrong.)

So the correct option for the first part is "The hypotenuse is not \(\sqrt{3}\) times longer than the longer leg."

Now, solving for \(x\) correctly:

In a right triangle, \(\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}\). Here, \(\theta = 30^\circ\), adjacent side (to \(30^\circ\)) is \(7\sqrt{10}\), hypotenuse is \(x\).

We know that \(\cos(30^\circ)=\frac{\sqrt{3}}{2}\).

So, \(\frac{\sqrt{3}}{2}=\frac{7\sqrt{10}}{x}\)

Cross - multiply: \(x\times\sqrt{3}=2\times7\sqrt{10}\)

\(x=\frac{14\sqrt{10}}{\sqrt{3}}\)

Rationalize the denominator by multiplying numerator and denominator by \(\sqrt{3}\):

\(x = \frac{14\sqrt{10}\times\sqrt{3}}{\sqrt{3}\times\sqrt{3}}=\frac{14\sqrt{30}}{3}\)