To compute the perimeter of the triangle formed by the points \( R(-4, 2) \), \( P(1, 2) \), and \( Q(1, 6) \), we need to find the lengths of the sides of the triangle using the distance formula. The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Step 1: Calculate the lengths of each side
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Length of \( RP \):
- \( R(-4, 2) \) and \( P(1, 2) \) \[ RP = \sqrt{(1 - (-4))^2 + (2 - 2)^2} = \sqrt{(1 + 4)^2 + 0^2} = \sqrt{5^2} = 5 \]
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Length of \( PQ \):
- \( P(1, 2) \) and \( Q(1, 6) \) \[ PQ = \sqrt{(1 - 1)^2 + (6 - 2)^2} = \sqrt{0^2 + (4)^2} = \sqrt{16} = 4 \]
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Length of \( QR \):
- \( Q(1, 6) \) and \( R(-4, 2) \) \[ QR = \sqrt{(-4 - 1)^2 + (2 - 6)^2} = \sqrt{(-5)^2 + (-4)^2} = \sqrt{25 + 16} = \sqrt{41} \]
Step 2: Compute the perimeter
Now we can find the perimeter of the triangle by summing up the lengths of its sides:
\[ \text{Perimeter} = RP + PQ + QR = 5 + 4 + \sqrt{41} \]
Calculating \(\sqrt{41}\):
\[ \sqrt{41} \approx 6.4 \quad (\text{rounded to the nearest tenth}) \]
Thus,
\[ \text{Perimeter} \approx 5 + 4 + 6.4 = 15.4 \]
Final result
The perimeter of the triangle, rounded to the nearest tenth, is
\[ \boxed{15.4} \]
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