Find the median through point A of the triangle ABC  with its sides equating 8,9,11 centimeters

To find the length of the median through point \( A \) in a triangle \( ABC \), where the side lengths are \( AB = 8 \, \text{cm} \), \( AC = 9 \, \text{cm} \), and \( BC = 11 \, \text{cm} \), we use the formula for the length of a median from a vertex in a triangle.

The formula for the median \( m_a \) from vertex \( A \) is given by:

\[
m_a = \frac{1}{2} \sqrt{2b^2 + 2c^2 - a^2}
\]

where:
- \( a \) is the side opposite to vertex \( A \) (which is \( BC = 11 \, \text{cm} \)),
- \( b \) is the side opposite to vertex \( B \) (which is \( AC = 9 \, \text{cm} \)),
- \( c \) is the side opposite to vertex \( C \) (which is \( AB = 8 \, \text{cm} \)).

Substituting these values into the formula:

\[
m_a = \frac{1}{2} \sqrt{2(9^2) + 2(8^2) - 11^2}
\]

Let's calculate this.The length of the median through point \( A \) of triangle \( ABC \) is \( 6.5 \, \text{cm} \).

Would you like more details on this solution or have any other questions? Here are five related questions to further explore:

1. How do you find the centroid of a triangle using medians?
2. How would the median formula change for an equilateral triangle?
3. What is the relationship between medians and altitudes in a triangle?
4. Can you derive the median formula from the Pythagorean theorem?
5. How do you calculate the area of a triangle using its medians?

**Tip**: The three medians of a triangle intersect at a single point called the centroid, which divides each median in a 2:1 ratio.

Find the median through point A of the triangle ABC with its sides equating 8,9,11 centimeters.

Discover how to find the median through point A in triangle ABC with sides measuring 8, 9, and 11 centimeters. The solution uses the median formula to compute the length, covering key geometric principles. Ideal for high school students in Grades 10-12.

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