The problem is a geometry problem. Pierre wants to calculate the distance between two buildings L and M situated on either side of a river. He measures the distance between M and an accessible point T to be $TM = 20m$. He measures the angles $\angle LTM = 68^{\circ}$ and $\angle MLT = 75^{\circ}$. The line MH is perpendicular to the line LT. We need to calculate the rounded to the nearest tenth of the lengths MH and LM.

1. Problem Description

The problem is a geometry problem. Pierre wants to calculate the distance between two buildings L and M situated on either side of a river. He measures the distance between M and an accessible point T to be $TM = 20m$. He measures the angles $\angle LTM = 68^{\circ}$ and $\angle MLT = 75^{\circ}$. The line MH is perpendicular to the line LT. We need to calculate the rounded to the nearest tenth of the lengths MH and LM.

2. Solution Steps

First, find the angle $\angle HMT$.

Since MH is perpendicular to LT, $\angle MHT = 90^{\circ}$. In triangle MHT, the sum of angles is $180^{\circ}$. We have $\angle MHT + \angle HTM + \angle HMT = 180^{\circ}$. Also, $\angle HTM = \angle LTM = 68^{\circ}$.

Therefore, $90^{\circ} + 68^{\circ} + \angle HMT = 180^{\circ}$, which means $\angle HMT = 180^{\circ} - 90^{\circ} - 68^{\circ} = 22^{\circ}$.

Now, let us find the length of MH.

In right triangle MHT, $\tan(\angle HTM) = \frac{MH}{TM}$.

$\tan(68^{\circ}) = \frac{MH}{20}$

$MH = 20 \tan(68^{\circ})$

$MH \approx 20 * 2.475 = 49.5$

Therefore, $MH \approx 49.5 m$

Next, let us find $\angle LMT$.

$\angle LMT = 180^{\circ} - \angle LTM - \angle MLT$ is wrong. We don't know what $\angle MLT$ is.

In triangle LMT, we are given $TM=20$, $\angle LTM = 68^\circ$ and $\angle MLT = 75^\circ$. The angle $\angle LMT$ is $180 - 68 - 75 = 37^\circ$.

By the Law of Sines, we have $\frac{LM}{\sin(\angle LTM)} = \frac{TM}{\sin(\angle MLT)}$ is wrong

By the Law of Sines, $\frac{LM}{\sin(\angle LTM)} = \frac{LT}{\sin(\angle LMT)} = \frac{MT}{\sin(\angle MLT)}$.

Therefore, $\frac{LM}{\sin(68^\circ)} = \frac{20}{\sin(75^\circ)}$.

$LM = \frac{20 * \sin(68^\circ)}{\sin(75^\circ)}$

$LM = \frac{20 * 0.927}{0.966} \approx \frac{18.54}{0.966} \approx 19.2 m$

Therefore, $LM \approx 19.2 m$

3. Final Answer

$MH \approx 49.5 m$

$LM \approx 19.2 m$