Math Problem Statement
Solution
The question provided is:
If , , and are the vertices of , then the length of the median through the vertex is?
Solution:
To solve for the length of the median from vertex , we follow these steps:
- Find the midpoint of side :
The coordinates of are , and are .
The formula for the midpoint of a line segment with endpoints and is:
Substitute the coordinates of and :
Thus, the midpoint of is .
- Calculate the length of the median :
The length of a line segment between two points and is given by:
Here, and . Substituting these into the formula:
Simplify step by step:
Now square these values:
Add these together:
Convert to have a denominator of :
Thus:
Simplify:
Final Answer:
The length of the median through vertex is:
Do you want me to explain any specific step further? Here are related questions you can explore:
- What is the formula for a median of a triangle?
- How do you find the centroid of a triangle using coordinates?
- How would the problem change if another vertex was introduced?
- What is the significance of a median in geometry?
- How would you calculate the perimeter of ?
Tip: The midpoint formula is essential in coordinate geometry for dividing a line segment into equal halves!
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Math Problem Analysis
Mathematical Concepts
Coordinate Geometry
Triangle Properties
Median of a Triangle
Formulas
Midpoint Formula: M = ((x1 + x2)/2, (y1 + y2)/2)
Distance Formula: d = √((x2 - x1)^2 + (y2 - y1)^2)
Theorems
Properties of Median in a Triangle
Suitable Grade Level
Grades 9-11
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