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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.

Question: In triangle ABC, AB = 7 cm, BC = 9 cm, and CA = 10 cm. What is the length of the median from vertex A to side BC?

Select the best response
Select the best response from the options.

Response 1 Response 2
Approximately 7.37 cm Approximately 6.84 cm
Response 3 Response 4
Approximately 8.12 cm Approximately 7.00 cm
Response 5 Response 6

Approximately 6.20 cm | don't have enough time to solve this question

Learn how to find the length of the median from vertex A to side BC in triangle ABC with side lengths AB = 7 cm, BC = 9 cm, and CA = 10 cm. This problem covers key geometric concepts and the formula for calculating the median in a triangle. Suitable for students in Grades 9-12.

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