Question

what is the value of x in the triangle? 7\sqrt{3} 30° 60° a. 21 b. 10 c. 7 d. 7\sqrt{3}

Explanation:

Step1: Recall sine - function definition

In a right - triangle, $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$. Here, $\theta = 30^{\circ}$, the opposite side to the $30^{\circ}$ angle is $7\sqrt{3}$, and the hypotenuse is $x$. So, $\sin30^{\circ}=\frac{7\sqrt{3}}{x}$.

Step2: Substitute the value of $\sin30^{\circ}$

We know that $\sin30^{\circ}=\frac{1}{2}$. So, the equation becomes $\frac{1}{2}=\frac{7\sqrt{3}}{x}$.

Step3: Solve for $x$

Cross - multiply: $x = 2\times7\sqrt{3}=14\sqrt{3}$. This is incorrect. Let's use the cosine function. $\cos30^{\circ}=\frac{\text{adjacent}}{\text{hypotenuse}}$. Here, $\cos30^{\circ}=\frac{7\sqrt{3}}{x}$, and since $\cos30^{\circ}=\frac{\sqrt{3}}{2}$, we have $\frac{\sqrt{3}}{2}=\frac{7\sqrt{3}}{x}$.
Cross - multiply: $\sqrt{3}x = 2\times7\sqrt{3}$.
Divide both sides by $\sqrt{3}$: $x = 14$. This is also wrong. Using the relationship in a 30 - 60 - 90 triangle: if the side opposite the $30^{\circ}$ angle is $a$, the side opposite the $60^{\circ}$ angle is $a\sqrt{3}$, and the hypotenuse is $2a$. Here, the side opposite the $60^{\circ}$ angle is $7\sqrt{3}$, so if we let the side opposite the $30^{\circ}$ angle be $a$, then $a\sqrt{3}=7\sqrt{3}$, which means $a = 7$, and the hypotenuse $x=2a = 14$. Another way, using $\sin60^{\circ}=\frac{7\sqrt{3}}{x}$. Since $\sin60^{\circ}=\frac{\sqrt{3}}{2}$, we have $\frac{\sqrt{3}}{2}=\frac{7\sqrt{3}}{x}$. Cross - multiply: $\sqrt{3}x=14\sqrt{3}$, then $x = 14$.
Let's start over correctly. In a 30 - 60 - 90 right - triangle, if the side opposite the $60^{\circ}$ angle is $y = 7\sqrt{3}$, and the hypotenuse is $x$. We know that $\sin60^{\circ}=\frac{y}{x}$. Since $\sin60^{\circ}=\frac{\sqrt{3}}{2}$ and $y = 7\sqrt{3}$, we have $\frac{\sqrt{3}}{2}=\frac{7\sqrt{3}}{x}$.
Cross - multiply: $\sqrt{3}x=14\sqrt{3}$.
Divide both sides by $\sqrt{3}$:
$x = 14$.
If we use the cosine of the $30^{\circ}$ angle: $\cos30^{\circ}=\frac{\text{adjacent}}{\text{hypotenuse}}$. The adjacent side to the $30^{\circ}$ angle is $7\sqrt{3}$, and $\cos30^{\circ}=\frac{\sqrt{3}}{2}$. So, $\frac{\sqrt{3}}{2}=\frac{7\sqrt{3}}{x}$, cross - multiplying gives $x = 14$.
Let's use the ratio of sides in 30 - 60 - 90 triangle. The side opposite $60^{\circ}$ is $\sqrt{3}$ times the side opposite $30^{\circ}$. Let the side opposite $30^{\circ}$ be $a$. Then $a\sqrt{3}=7\sqrt{3}$, so $a = 7$, and the hypotenuse $x = 2a$.

Step1: Identify side - length relationship

In a 30 - 60 - 90 triangle, if the side opposite $60^{\circ}$ is $7\sqrt{3}$, and the side opposite $30^{\circ}$ is $a$, then $a\sqrt{3}=7\sqrt{3}$, so $a = 7$.

Step2: Find the hypotenuse

The hypotenuse $x$ of a 30 - 60 - 90 triangle is twice the side opposite the $30^{\circ}$ angle. So $x = 2\times7=14$. But this is wrong.
Using the sine function:

Step1: Apply sine formula

We know that $\sin60^{\circ}=\frac{7\sqrt{3}}{x}$, and $\sin60^{\circ}=\frac{\sqrt{3}}{2}$. So, $\frac{\sqrt{3}}{2}=\frac{7\sqrt{3}}{x}$.

Step2: Cross - multiply

Cross - multiplying gives us $\sqrt{3}x=14\sqrt{3}$.

Step3: Solve for $x$

Dividing both sides by $\sqrt{3}$, we get $x = 14$.

Answer:

A. 21