​ geometry problem with triangles

To solve this problem, we need to understand the relationships within the triangle STU. Here are the steps to find the length of $SX$:

 

### Step 1: Analyzing the Triangle

Given:

- $S$, $T$, and $U$ are the vertices of the triangle.

- $M$ is the midpoint of $ST$.

- $N$ is a point on $TU$ such that $SN$ is the altitude of the triangle.

- $ST = SU = 13$, $TU = 8$.

- $UM$ and $SN$ intersect at $X$.

 

### Step 2: Applying the Median and Altitude Properties

Since $M$ is the midpoint of $ST$, $SM = MT = \frac{13}{2} = 6.5$.

 

Also, $SN$ is an altitude, so it is perpendicular to $TU$.

 

### Step 3: Use the Property of the Centroid

In any triangle, the centroid (intersection of the medians) divides each median in a 2:1 ratio. Since $X$ is the intersection of the medians $SN$ and $UM$, it is the centroid of triangle $STU$.

 

This implies:

$$SX = \frac{2}{3} \times SN$$

where $SN$ is the altitude from $S$ to $TU$.

 

### Step 4: Calculate SN Using the Area of the Triangle

We use the fact that the area of the triangle can be calculated in two ways:

1. Using base $TU$ and height $SN$.

2. Using Heron's formula.

 

#### Heron's Formula:

First, calculate the semi-perimeter $s$:

$$s = \frac{ST + SU + TU}{2} = \frac{13 + 13 + 8}{2} = 17$$

Then, calculate the area $\Delta$:

$$\Delta = \sqrt{s(s - ST)(s - SU)(s - TU)} = \sqrt{17(17 - 13)(17 - 13)(17 - 8)} = \sqrt{17 \times 4 \times 4 \times 9} = \sqrt{2448} = 24$$

 

#### Area Using Altitude $SN$:

The area can also be written as:

$$\Delta = \frac{1}{2} \times TU \times SN = \frac{1}{2} \times 8 \times SN = 4 \times SN$$

Equating the two expressions for the area:

$$24 = 4 \times SN \implies SN = 6$$

 

### Step 5: Calculate SX

Now that we know $SN = 6$, the length of $SX$ is:

$$SX = \frac{2}{3} \times 6 = 4$$

 

Thus, the length of $SX$ is $\boxed{4}$.