Point $G$ is the midpoint of median $\overline{XM}$ of $\triangle XYZ$. Point $H$ is the midpoint of $\overline{XY}$, and point $T$ is the intersection of $\overline{HM}$ and $\overline{YG}$. Find the area of $\triangle MTG$ if $[XYZ] =150$.
To find the area of triangle $MTG$, we first need to find the lengths of the sides of triangle $MTG$. Let's go step by step:
1. Recognize that triangle $XYZ$ is not directly relevant to finding the area of triangle $MTG$. We only need the fact that its area is given as $[XYZ] = 150$ as given in the problem.
2. Since $G$ is the midpoint of median $\overline{XM}$, we can conclude that $\overline{XG} = \overline{GM}$. Similarly, since $H$ is the midpoint of $\overline{XY}$, we have $\overline{XH} = \overline{HY}$. These midpoint properties will be helpful in finding the lengths of $\overline{YM}$ and $\overline{YH}$.
3. Let's start by finding the length of $\overline{YM}$. Since $G$ is the midpoint of $\overline{XM}$, we have $\overline{XM} = 2 \cdot \overline{XG}$. We can also observe that $\overline{XM} = \overline{XY} + \overline{YM}$. Combining these two equations, we get $\overline{XY} + \overline{YM} = 2 \cdot \overline{XG}$. Since we know $\overline{XY} = 2 \cdot \overline{XH}$, we can substitute this value into the equation: $2 \cdot \overline{XH} + \overline{YM} = 2 \cdot \overline{XG}$. Rearranging, we find $\overline{YM} = 2 \cdot \overline{XG} - 2 \cdot \overline{XH} = 2(\overline{XG} - \overline{XH})$.
4. Now, let's find the length of $\overline{YH}$. From earlier, we know $\overline{XH} = \overline{HY}$. Since $H$ is the midpoint of $\overline{XY}$, we have $\overline{XY} = 2 \overline{XH}$. Substituting this into the equation, we get $\overline{XY} = 2 \cdot \overline{XH} = 2 \cdot \overline{HY}$. Therefore, $\overline{YH} = \frac{1}{2} \cdot \overline{XY}$.
5. Now that we have the lengths of $\overline{YM}$ and $\overline{YH}$, we can find the length of $\overline{MG}$. Since $G$ is the midpoint of $\overline{XM}$, we have $\overline{XG} = \frac{1}{2} \cdot \overline{XM}$. Substituting the value of $\overline{YM}$ that we found earlier, we get $\overline{XG} = \frac{1}{2} \cdot (2(\overline{XG} - \overline{XH})) = \overline{XG} - \overline{XH}$. This means that $\overline{MG} = \overline{XH}$.
6. To find the length of $\overline{MT}$, we can use a similar approach. Since $T$ is the intersection of $\overline{HM}$ and $\overline{YG}$, we have $\overline{HM} = \overline{MT}$. Now, we need to find the length of $\overline{HY}$. From earlier, we know $\overline{YH} = \frac{1}{2} \cdot \overline{XY}$. Since $\overline{XY} = 2 \cdot \overline{XH}$, we can substitute this value to get $\overline{YH} = \frac{1}{2} \cdot 2 \cdot \overline{XH} = \overline{XH}$. Therefore, $\overline{HY} = \overline{XH}$. This means that $\overline{MT} = \overline{HM} = \overline{HY}$.
7. Now that we have the lengths of all three sides of triangle $MTG$, we can find its area using Heron's formula or simply multiplying the base $\overline{MT}$ by the height $\overline{TG}$. Since $\overline{TG}$ is perpendicular to $\overline{MT}$, it is also the height of the triangle. Therefore, the area of triangle $MTG$ is given by $\frac{1}{2} \cdot \overline{MT} \cdot \overline{TG}$.
By following these steps, you should now have all the information needed to find the area of triangle $MTG$.