The problem asks to find the tangent of the two acute angles, $M$ and $N$, in a given right triangle $MNL$. The lengths of the sides are given as $ML = 8$, $NL = 6$, and $MN = 10$. We need to express the tangents as both a fraction and a decimal rounded to 4 decimal places.

1. Problem Description

The problem asks to find the tangent of the two acute angles, $M$ and $N$, in a given right triangle $MNL$. The lengths of the sides are given as $ML = 8$, $NL = 6$, and $MN = 10$. We need to express the tangents as both a fraction and a decimal rounded to 4 decimal places.

2. Solution Steps

The tangent of an angle in a right triangle is defined as the ratio of the length of the side opposite to the angle to the length of the side adjacent to the angle.

For angle $M$:

The side opposite to $M$ is $NL$, which has a length of

6. The side adjacent to $M$ is $ML$, which has a length of

8. Therefore,

$tan(M) = \frac{NL}{ML} = \frac{6}{8} = \frac{3}{4}$

To convert this fraction to a decimal, we divide 3 by 4:

$tan(M) = \frac{3}{4} = 0.7500$

For angle $N$:

The side opposite to $N$ is $ML$, which has a length of

8. The side adjacent to $N$ is $NL$, which has a length of

6. Therefore,

$tan(N) = \frac{ML}{NL} = \frac{8}{6} = \frac{4}{3}$

To convert this fraction to a decimal, we divide 4 by 3:

$tan(N) = \frac{4}{3} = 1.333333...$

Rounding to 4 decimal places, we have:

$tan(N) \approx 1.3333$

3. Final Answer

$tan M = \frac{3}{4} = 0.7500$

$tan N = \frac{4}{3} \approx 1.3333$