The sides of the triangle are 5, 8, 10 respectively. Find the lengths of the medians.
The correct answer and explanation is :
Given a triangle with sides 5, 8, and 10, we can calculate the lengths of the medians using Apollonius’s theorem or formula for the medians.
Step 1: Understanding Medians
A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side. The three medians of a triangle are typically denoted as ( m_a ), ( m_b ), and ( m_c ), corresponding to the sides ( a ), ( b ), and ( c ) respectively.
The length of a median from a vertex opposite to side ( a ) is given by the formula:
[
m_a = \sqrt{\frac{2b^2 + 2c^2 – a^2}{4}}
]
where:
- ( a ), ( b ), and ( c ) are the lengths of the sides of the triangle.
In this case, the sides are 5, 8, and 10. So, we have ( a = 5 ), ( b = 8 ), and ( c = 10 ).
Step 2: Calculate the Medians
Median ( m_a ) (opposite side 5):
[
m_a = \sqrt{\frac{2(8^2) + 2(10^2) – 5^2}{4}} = \sqrt{\frac{2(64) + 2(100) – 25}{4}} = \sqrt{\frac{128 + 200 – 25}{4}} = \sqrt{\frac{303}{4}} = \sqrt{75.75} \approx 8.69
]
Median ( m_b ) (opposite side 8):
[
m_b = \sqrt{\frac{2(5^2) + 2(10^2) – 8^2}{4}} = \sqrt{\frac{2(25) + 2(100) – 64}{4}} = \sqrt{\frac{50 + 200 – 64}{4}} = \sqrt{\frac{186}{4}} = \sqrt{46.5} \approx 6.82
]
Median ( m_c ) (opposite side 10):
[
m_c = \sqrt{\frac{2(5^2) + 2(8^2) – 10^2}{4}} = \sqrt{\frac{2(25) + 2(64) – 100}{4}} = \sqrt{\frac{50 + 128 – 100}{4}} = \sqrt{\frac{78}{4}} = \sqrt{19.5} \approx 4.41
]
Step 3: Final Results
The lengths of the medians are approximately:
- Median ( m_a ) ≈ 8.69
- Median ( m_b ) ≈ 6.82
- Median ( m_c ) ≈ 4.41
Conclusion
To summarize, for a triangle with sides 5, 8, and 10, the medians have the following lengths:
- ( m_a \approx 8.69 )
- ( m_b \approx 6.82 )
- ( m_c \approx 4.41 )
These results were derived using the median length formula, which depends on the side lengths of the triangle.