We are tasked with finding the length of the median from vertex $A$ to side $BC$ in triangle $ABC$, where:

• $AB = 7 \, \text{cm}$
• $BC = 9 \, \text{cm}$
• $CA = 10 \, \text{cm}$

The formula for the length of the median from vertex $A$ to side $BC$ is given by the following expression:

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

where:

• $a = BC$
• $b = AC$
• $c = AB$

Let's substitute the values into the formula and calculate.

• $a = 9 \, \text{cm}$
• $b = 10 \, \text{cm}$
• $c = 7 \, \text{cm}$

Now, I'll perform the calculation.The length of the median from vertex $A$ to side $BC$ is approximately $7.37 \, \text{cm}$.

Thus, the correct response is Response 1: Approximately 7.37 cm.

Would you like further details on this calculation or any other questions related to it?

Here are five related questions:

1. How do you derive the formula for the median in a triangle?
2. How would the result change if the side lengths were different?
3. What is the geometric significance of a median in a triangle?
4. How do you find the centroid of a triangle, and how does it relate to the medians?
5. Can the length of a median ever exceed the length of the triangle's longest side?

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