The vertices of triangle PQR are P(2,5),Q(6,1)and R(-2,-1).Find the length of the longest median.
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To find the length of the longest median of triangle PQR, we first need to calculate the lengths of all three medians and then determine which one is the longest.
Step 1: Find the midpoint of each side of the triangle.
The midpoint of a line segment with endpoints (x₁, y₁) and (x₂, y₂) is given by the formula:
Midpoint = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)
Let's find the midpoints of each side of the triangle PQR:
Midpoint of PQ:
x-coordinate: (2 + 6) / 2 = 4
y-coordinate: (5 + 1) / 2 = 3
Midpoint of PQ = (4, 3)
Midpoint of QR:
x-coordinate: (6 - 2) / 2 = 2
y-coordinate: (1 - (-1)) / 2 = 1
Midpoint of QR = (2, 1)
Midpoint of RP:
x-coordinate: (2 + (-2)) / 2 = 0
y-coordinate: (5 + (-1)) / 2 = 2
Midpoint of RP = (0, 2)
Step 2: Calculate the lengths of the medians.
The median of a triangle is a line segment that connects a vertex with the midpoint of the opposite side. We need to find the lengths of the medians PR, PQ, and QR.
Length of PR = Distance between points P(2,5) and (0,2)
Using the distance formula:
Distance = √((x₂ - x₁)² + (y₂ - y₁)²)
Distance PR = √((0 - 2)² + (2 - 5)²) = √((-2)² + (-3)²) = √(4 + 9) = √13
Length of PQ = Distance between points P(2,5) and (4,3)
Distance PQ = √((4 - 2)² + (3 - 5)²) = √(2² + (-2)²) = √(4 + 4) = √8 = 2√2
Length of QR= Distance between points Q(6,1) and (2,1)
Distance QR = √((2 - 6)² + (1 - 1)²) = √((-4)² + 0²) = √(16 + 0) = √16 = 4
Step 3: Determine the longest median.
Compare the lengths of the three medians: √13, 2√2, and 4. The longest median is 4 units, which corresponds to the median QR.
Therefore, the length of the longest median of triangle PQR is 4 units.