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 $[XYZ] =150$.

wtf?

To find the area of triangle $MTG$, we need to determine the lengths of the sides $MT$, $TG$, and $MG$. Once we have these lengths, we can use the formula for the area of a triangle given its side lengths, known as Heron's formula.

Let's break down the problem step by step:

1. Identify the given information: We are given that $G$ is the midpoint of median $\overline{XM}$, $H$ is the midpoint of $\overline{XY}$, and $T$ is the intersection of $\overline{HM}$ and $\overline{YG}$. We are also told that the area of $\triangle XYZ$ is 150.

2. Understand the properties of a median: In a triangle, a median is a line segment that connects a vertex to the midpoint of the opposite side. In this case, $GM$ is a median of $\triangle XYZ$, and $G$ is the midpoint of $XM$. This means that $XM = 2GM$, or equivalently, $GM = \frac{1}{2}XM$.

3. Determine the length of $GM$: Since $G$ is the midpoint of $XM$, this means that $GM$ is half the length of $XM$. So, let's call $XM$ as $2x$, and then $GM = \frac{1}{2}(2x) = x$.

4. Identify the lengths of other line segments: Given that $H$ is the midpoint of $\overline{XY}$, this means that $XH = HY$, and $HM$ is half the length of $XY$, so $HM = \frac{1}{2}XY$. Also, $YT = TG$ because $YG$ is a median of $\triangle XYZ$, and $T$ is the midpoint of $\overline{YG}$.

5. Determine the length of $HM$: Since $HM$ is half the length of $XY$, let's call $XY$ as $2a$, and then $HM = \frac{1}{2}(2a) = a$.

6. Determine the length of $TG$: Since $YT = TG$, and we know $HM = a$, and $HY = XH = a$, we can use the fact that triangles $\triangle MHT$ and $\triangle HYT$ are similar by the SAS (side-angle-side) similarity criterion. Therefore, $\frac{TG}{YT} = \frac{HM}{HY}$. Plugging in these values, we get $\frac{TG}{TG} = \frac{a}{a}$, which simplifies to $1 = 1$. So, $TG = YT = a$.

7. Determine the length of $MT$: We know that $XM = 2x$, and $HG = a$. Therefore, $MT = XM - XH - HG = 2x - a - a = 2x - 2a$.

8. Calculate the area of triangle $MTG$: Now that we have the lengths of the sides $MT$, $TG$, and $MG$, we can use Heron's formula to calculate the area. Heron's formula states that the area of a triangle with side lengths $a$, $b$, and $c$ is given by the formula:

\[ \text{Area} = \sqrt{s(s-a)(s-b)(s-c)} \]

where $s$ is the semiperimeter of the triangle, defined as:

\[ s = \frac{a+b+c}{2} \]

Plugging in the values, we have $a = 2x - 2a$, $b = a$, and $c = x$. The semiperimeter $s$ is then $s = \frac{a + b + c}{2} = \frac{(2x - 2a) + a + x}{2} = \frac{3x - a}{2}$.

Substituting the values into Heron's formula, we get:

\[ \text{Area} = \sqrt{\left(\frac{3x - a}{2}\right)\left(\frac{3x - a}{2} - (2x - 2a)\right)\left(\frac{3x - a}{2} - a\right)\left(\frac{3x - a}{2} - x\right)} \]

Simplifying this expression will provide the area of triangle $MTG$.

Please note that we'll need additional information to determine the values of $x$ and $a$, which can be found by using properties of medians or by utilizing any additional information given in the original problem statement.