To find the perimeter of the triangle formed by the points \( A(1, 5) \), \( B(4, 1) \), and \( C(-4, -5) \), we will follow these steps:
Step 1: Calculate the Length of Each Side
We will use the distance formula to find the lengths of each side of the triangle. The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Calculate Length of Side \( AB \)
Points \( A(1, 5) \) and \( B(4, 1) \): \[ AB = \sqrt{(4 - 1)^2 + (1 - 5)^2} \] \[ = \sqrt{(3)^2 + (-4)^2} \] \[ = \sqrt{9 + 16} \] \[ = \sqrt{25} = 5 \]
Calculate Length of Side \( BC \)
Points \( B(4, 1) \) and \( C(-4, -5) \): \[ BC = \sqrt{(-4 - 4)^2 + (-5 - 1)^2} \] \[ = \sqrt{(-8)^2 + (-6)^2} \] \[ = \sqrt{64 + 36} \] \[ = \sqrt{100} = 10 \]
Calculate Length of Side \( CA \)
Points \( C(-4, -5) \) and \( A(1, 5) \): \[ CA = \sqrt{(1 - (-4))^2 + (5 - (-5))^2} \] \[ = \sqrt{(1 + 4)^2 + (5 + 5)^2} \] \[ = \sqrt{(5)^2 + (10)^2} \] \[ = \sqrt{25 + 100} \] \[ = \sqrt{125} = 5\sqrt{5} \approx 11.2 \]
Step 2: Calculate the Perimeter
The perimeter \( P \) of the triangle is the sum of the lengths of its sides:
\[ P = AB + BC + CA \] \[ = 5 + 10 + 5\sqrt{5} \] Using \( \sqrt{5} \approx 2.236 \):
\[ 5\sqrt{5} \approx 5 \times 2.236 \approx 11.18 \]
Now we can approximate the perimeter:
\[ P \approx 5 + 10 + 11.18 \approx 26.18 \]
Final Step: Round to the Nearest Tenth
Rounding \( 26.18 \) to the nearest tenth gives:
\[ P \approx 26.2 \]
Conclusion
The perimeter of the triangle, rounded to the nearest tenth of a unit, is 26.2.
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