Question

1. find the length of the missing side. round all answers to two decimal places.

Explanation:

Step1: Recall tangent formula for right - triangle

In a right - triangle, $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$.

Step2: Solve for the first triangle

For the triangle with $\theta = 25^{\circ}$ and adjacent side $a = 10$, we want to find the opposite side $x$. Using $\tan\theta=\frac{x}{a}$, so $x = a\times\tan\theta$. Substituting $\theta = 25^{\circ}$ and $a = 10$, we get $x=10\times\tan(25^{\circ})$. Since $\tan(25^{\circ})\approx0.4663$, then $x\approx10\times0.4663 = 4.66$.

Step3: Solve for the second triangle

For the triangle with $\theta = 48^{\circ}$ and adjacent side $a = 8$, using $\tan\theta=\frac{x}{a}$, we have $x=a\times\tan\theta$. Substituting $\theta = 48^{\circ}$ and $a = 8$, and since $\tan(48^{\circ})\approx1.1106$, then $x\approx8\times1.1106 = 8.88$.

Step4: Solve for the third triangle

For the triangle with $\theta = 54^{\circ}$ and adjacent side $a = x$ and opposite side $o = 12$. Using $\tan\theta=\frac{o}{a}$, we can rewrite it as $a=\frac{o}{\tan\theta}$. Substituting $\theta = 54^{\circ}$ and $o = 12$, and since $\tan(54^{\circ})\approx1.3764$, then $x=\frac{12}{\tan(54^{\circ})}\approx\frac{12}{1.3764}\approx8.71$.

Answer:

The lengths of $x$ for the three triangles are approximately $4.66$, $8.88$, and $8.71$ respectively.