The vertices of a triangle are A(5,−1), B(−3,−2), and C(1,8). Find the length of the median drawn from vertex A to side BC, and determine the coordinates of the centroid of △ABC.
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Step-by-Step Solution
Step 1: Find the midpoint of BC
To find the median from vertex A to side BC, we first need to find the midpoint of side BC. Let's call this midpoint D. We use the midpoint formula, which averages the x-coordinates and y-coordinates of points B and C.
Step 2: Calculate coordinates of D
Substitute the coordinates of B (−3,−2) and C (1,8) into the midpoint formula. This gives us the coordinates of point D, which is (−1,3).
Step 3: Calculate the length of the median AD
Now that we have the coordinates of A (5,−1) and D (−1,3), we can find the length of the median AD using the distance formula. The distance formula calculates the length of a line segment between two points.
Step 4: Calculate length of AD
Substitute the coordinates of A (5,−1) and D (−1,3) into the distance formula. After performing the calculations, we find the length of the median AD to be 52, which simplifies to 213 units.
Step 5: Find the coordinates of the centroid
The centroid of a triangle is the point where the three medians intersect. Its coordinates are found by averaging the x-coordinates and y-coordinates of all three vertices of the triangle.
Step 6: Calculate coordinates of the centroid G
Substitute the coordinates of A (5,−1), B (−3,−2), and C (1,8) into the centroid formula. After summing the coordinates and dividing by 3, we find the centroid G to be (1,35).