​ 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 \( \boxed{4} \).