To find the perimeter of the triangle formed by the points \( A(1, 5) \), \( B(4, 1) \), and \( C(-4, -5) \), we need to calculate the lengths of the sides of the triangle and then sum those lengths.
Step 1: Calculate the lengths of each side
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Length of side AB: The distance \( d \) between points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) can be calculated using the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] For the points \( A(1, 5) \) and \( B(4, 1) \): \[ d_{AB} = \sqrt{(4 - 1)^2 + (1 - 5)^2} = \sqrt{(3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
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Length of side BC: For points \( B(4, 1) \) and \( C(-4, -5) \): \[ d_{BC} = \sqrt{(-4 - 4)^2 + (-5 - 1)^2} = \sqrt{(-8)^2 + (-6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \]
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Length of side CA: For points \( C(-4, -5) \) and \( A(1, 5) \): \[ d_{CA} = \sqrt{(1 - (-4))^2 + (5 - (-5))^2} = \sqrt{(1 + 4)^2 + (5 + 5)^2} = \sqrt{5^2 + 10^2} = \sqrt{25 + 100} = \sqrt{125} = 5\sqrt{5} \]
Step 2: Calculate the perimeter
Now, we can find the perimeter \( P \) of the triangle by adding the lengths of the three sides: \[ P = d_{AB} + d_{BC} + d_{CA} = 5 + 10 + 5\sqrt{5} \]
Step 3: Approximate the perimeter
To find the numerical value of the perimeter, we'll evaluate \( 5\sqrt{5} \) and add it to 15.
First, calculate \( \sqrt{5} \): \[ \sqrt{5} \approx 2.236 \]
Then, \[ 5\sqrt{5} \approx 5 \times 2.236 \approx 11.18 \]
Now add this to the other lengths: \[ P \approx 5 + 10 + 11.18 = 26.18 \]
Step 4: Round to the nearest tenth
Finally, rounding \( 26.18 \) to the nearest tenth gives us: \[ \text{Perimeter} \approx 26.2 \]
Conclusion
Thus, the perimeter of the triangle, rounded to the nearest tenth of a unit, is \( \boxed{26.2} \).
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