Median from the Right Angle to the Hypotenuse

The **median of a triangle** is a line segment that joins a vertex to the midpoint of the opposite side. In a right-angled triangle, one important case is when the median is drawn from the right-angled vertex to the hypotenuse. Let's go over some theorems and examples related to this concept.

### 1. **Theorem: Median from the Right Angle to the Hypotenuse**

In a right-angled triangle, the median drawn from the right-angled vertex to the hypotenuse is half the length of the hypotenuse.

#### Proof:

Consider a right-angled triangle \(ABC\) where \(\angle ABC = 90^\circ\), and let \(M\) be the midpoint of the hypotenuse \(AC\). We want to show that the length of median \(BM\) is half of \(AC\).

1. Since \(M\) is the midpoint of \(AC\), \(AM = MC\).
2. Triangles \(ABM\) and \(CBM\) are congruent by the RHS (Right angle-Hypotenuse-Side) criterion:
- \(AB = BC\) (since both are legs of the right triangle),
- \(BM\) is common,
- \(\angle ABM = \angle CBM = 90^\circ\).

3. As a result, triangles \(ABM\) and \(CBM\) are congruent, which implies:
- \(AM = MC\)
- \(BM\) is perpendicular to \(AC\).

4. Thus, triangle \(BMC\) is a right-angled triangle with hypotenuse \(AC\). The length of median \(BM\) from vertex \(B\) to hypotenuse \(AC\) is exactly half of \(AC\). This is a special property unique to right-angled triangles.

Therefore, the median from the right angle to the hypotenuse in a right-angled triangle is half the length of the hypotenuse.

### 2. **Example**

Consider a right-angled triangle with sides \(AB = 3\), \(BC = 4\), and \(AC = 5\).

- The hypotenuse \(AC = 5\).
- The midpoint of \(AC\) is point \(M\).
- The length of the median \(BM\) from the right angle \(B\) to \(M\) is:

\[
BM = \frac{1}{2} \times AC = \frac{1}{2} \times 5 = 2.5.
\]

### 3. **Properties of the Median in Right-Angled Triangles**

1. The median drawn to the hypotenuse is always half the hypotenuse's length.
2. In general triangles (non-right-angled), the median does not necessarily relate to the sides in such a simple way.
3. Medians in triangles always intersect at a single point called the **centroid**, which divides each median into a 2:1 ratio.

Would you like more examples or any further explanations?