Question

in which triangle is the value of x equal to tan⁻¹\\(left\frac{3.1}{5.2}\
ight\\)? (images may not be drawn to scale.)

Explanation:

Step1: Recall Tangent Definition

In a right triangle, $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$ to angle $\theta$. We need to find which triangle has $\tan(x) = \frac{3.1}{5.2}$.

Step2: Analyze Each Triangle

• First Triangle: Opposite to $x$ would be the side not adjacent, adjacent is $3.1$, hypotenuse $5.2$? No, wait, right triangle: legs are $3.1$ (adjacent) and let's see, the other leg? Wait, no, $\tan(x)$: for angle $x$, adjacent and opposite. Wait, the third triangle (right - angled, legs $3.1$ and $5.2$? Wait, no, let's check the third triangle: right angle, one leg $3.1$, one leg $5.2$? Wait, no, the third triangle (the right - most one) has adjacent side (to $x$) as $5.2$? No, wait the first triangle: right angle, one leg $3.1$, hypotenuse $5.2$? No, the third triangle: right angle, one leg $3.1$, one leg $5.2$? Wait, no, the correct triangle is the one where the opposite side to $x$ is $3.1$ and adjacent is $5.2$? Wait, no, $\tan^{-1}(\frac{3.1}{5.2})$ means $\tan(x)=\frac{3.1}{5.2}$, so opposite over adjacent is $\frac{3.1}{5.2}$. So we need a right triangle where for angle $x$, opposite side is $3.1$ and adjacent is $5.2$, or vice - versa? Wait, no, $\tan(x)=\frac{\text{opposite}}{\text{adjacent}}$. So if $\tan(x)=\frac{3.1}{5.2}$, then opposite is $3.1$, adjacent is $5.2$. Looking at the triangles, the third triangle (the right - angled triangle with legs $3.1$ and $5.2$, where angle $x$ has opposite $3.1$ and adjacent $5.2$? Wait, no, the first triangle: leg $3.1$ (adjacent), hypotenuse $5.2$? No, the third triangle: right angle, one leg $3.1$, one leg $5.2$, and angle $x$ such that opposite is $3.1$ and adjacent is $5.2$. Wait, the third triangle (the right - most one) has the right angle, one leg $3.1$ (vertical), one leg $5.2$ (horizontal), and angle $x$ at the top. So for angle $x$, opposite side is $5.2$? No, wait no. Wait the first triangle: right angle, leg $3.1$ (horizontal), hypotenuse $5.2$, and angle $x$ at the top. Then opposite to $x$ is $3.1$, adjacent is the other leg. Wait, maybe I made a mistake. Wait the problem is $\tan^{-1}(\frac{3.1}{5.2})$, so $\tan(x)=\frac{3.1}{5.2}$. So in a right triangle, $\tan(x)=\frac{\text{opposite}}{\text{adjacent}}=\frac{3.1}{5.2}$. So we need to find the triangle where the opposite side to $x$ is $3.1$ and adjacent is $5.2$. Looking at the three triangles, the third triangle (the right - angled triangle with the horizontal leg $5.2$ and vertical leg $3.1$, and angle $x$ at the top) has opposite side (to $x$) as $5.2$? No, wait no. Wait the first triangle: right angle, horizontal leg $3.1$, hypotenuse $5.2$, angle $x$ at the top. Then opposite to $x$ is $3.1$, adjacent is the other leg. But we need $\frac{3.1}{5.2}$. Wait, maybe the third triangle: right angle, vertical leg $3.1$, horizontal leg $5.2$, angle $x$ at the top. Then $\tan(x)=\frac{\text{opposite}}{\text{adjacent}}=\frac{3.1}{5.2}$ (opposite is $3.1$, adjacent is $5.2$). So the third triangle (the right - most one) is the correct one.

Answer:

The Right - most Triangle (the third triangle in the given set of triangles)