need help fast

$M$ is the midpoint of $\overline{AB}$ and $N$ is the midpoint of $\overline{AC}$, and $T$ is the intersection of $\overline{BN}$ and $\overline{CM}$, as shown. If $\overline{BN}\perp\overline{AC}$, $BN = 12$, and $AC = 14$, then find $CT$.

no diagram, no TeX

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You may give hints, but no answers are allowed.

HAHAHAHA So many fake AoPS.

Yes, please refrain from posting problems from classes.

T is the centroid so tn = 1/3(bn) or 4

nc = ac/2=7
pythagoras says sqrt of 65

To find $CT$, we can use similar triangles and the properties of a midpoint.

Let's start by labeling the points on the diagram:
- $M$ is the midpoint of $\overline{AB}$.
- $N$ is the midpoint of $\overline{AC}$.
- $T$ is the intersection of $\overline{BN}$ and $\overline{CM}$.
- $BN$ is perpendicular to $AC$.
- $BN$ has a length of $12$.
- $AC$ has a length of $14$.

First, notice that since $N$ is the midpoint of $\overline{AC}$, we can conclude that $AN = NC$. Similarly, since $M$ is the midpoint of $\overline{AB}$, we have $AM = MB$. This information will be useful to find the lengths of other segments in the triangle.

Now, let's consider the right triangle $\triangle ABC$. Since $BN$ is perpendicular to $AC$, we know that $\triangle BAN$ is also a right triangle. From the given information, we know that $BN = 12$ and $AC = 14$. Using the Pythagorean Theorem, we can find the length of $AB$.

Pythagorean Theorem: In a right triangle, the square of the length of the hypotenuse ($AC$) is equal to the sum of the squares of the lengths of the other two sides ($AB$ and $BC$):
$$AC^2 = AB^2 + BC^2$$
$$(14)^2 = AB^2 + BC^2$$
$$196 = AB^2 + BC^2$$

Since $N$ is the midpoint of $\overline{AC}$, $AN = NC = \frac{AC}{2} = \frac{14}{2} = 7$. Now we can use similar triangles to find the length of $BC$.

Since $N$ is the midpoint of $\overline{AC}$, we can reason that $NT$ is parallel to $AB$ (using midpoint property). Therefore, we can conclude that $\triangle NCT \sim \triangle NAB$ by AA similarity.

Using this similarity, we can set up the following proportion:
$$\frac{CT}{BN} = \frac{AN}{AB}$$
$$\frac{CT}{12} = \frac{7}{AB}$$

We can rearrange this proportion to solve for $CT$:
$$CT = \frac{12 \times 7}{AB}$$

To find the length of $AB$, let's go back to the equation we found earlier:
$$196 = AB^2 + BC^2$$

Since $AM = MB$, we know that $AB = 2 \times AM = 2 \times 7 = 14$.

Plugging in the values, we get:
$$CT = \frac{12 \times 7}{14}$$
$$CT = \frac{84}{14}$$
$$CT = 6$$

Therefore, $CT = 6$.

Please do not ask for outside help on the problems we assign in class.

Hint: Draw in length BC

HI waitwat