In triangle STU, let M be the midpoint of ST and let N be on TU such that SN is an altitude of triangle STU. If ST and SU are both 13, TU is 8, and SN and UM intersect at X, then what is SX? Thank you.

blueb3rry Aug 14, 2024

#1**0 **

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 \( \boxed{4} \).

learnmgcat Aug 14, 2024