Question

calculate the lengths of the 2 unlabeled sides. give the exact answers.
type the answers in the boxes below.
mn = units
ln = units

Explanation:

Step1: Identify the triangle type

This is a right - triangle with one angle $60^{\circ}$ and the right - angle at $L$. So, the third angle at $N$ is $180^{\circ}-90^{\circ}-60^{\circ}=30^{\circ}$. In a $30 - 60 - 90$ right - triangle, the sides are in the ratio $1:\sqrt{3}:2$, where the side opposite the $30^{\circ}$ angle is the shortest side (let's call it $x$), the side opposite the $60^{\circ}$ angle is $x\sqrt{3}$, and the hypotenuse is $2x$.

In $\triangle LMN$, $\angle M = 60^{\circ}$, $\angle L=90^{\circ}$, so the side $LM = 2.5$ is adjacent to the $60^{\circ}$ angle and is the shorter leg (opposite the $30^{\circ}$ angle? Wait, no. Wait, $\angle N=30^{\circ}$, so the side opposite $\angle N$ is $LM = 2.5$. So, if the side opposite $30^{\circ}$ (i.e., $LM$) is $x = 2.5$, then the hypotenuse $MN$ (opposite the right - angle) is $2x$, and the side $LN$ (opposite the $60^{\circ}$ angle) is $x\sqrt{3}$.

Step2: Calculate the length of $MN$

Since in a $30 - 60 - 90$ triangle, the hypotenuse $c = 2\times$ (the side opposite $30^{\circ}$). The side opposite $30^{\circ}$ is $LM = 2.5$. So, $MN=2\times LM$.
$MN = 2\times2.5=5$

Step3: Calculate the length of $LN$

The side $LN$ is opposite the $60^{\circ}$ angle. In a $30 - 60 - 90$ triangle, the side opposite $60^{\circ}$ is $\sqrt{3}\times$ (the side opposite $30^{\circ}$). The side opposite $30^{\circ}$ is $LM = 2.5$. So, $LN=\sqrt{3}\times LM$.
$LN = 2.5\sqrt{3}=\frac{5\sqrt{3}}{2}$

Answer:

$MN = \boldsymbol{5}$ units
$LN=\boldsymbol{\frac{5\sqrt{3}}{2}}$ units