To find the perimeter of the triangle formed by the points (1, 5), (4, 1), and (−4, −5), Nathan needs to calculate the lengths of each of the triangle's sides using the distance formula, and then add those lengths together. Here’s how he can do that step-by-step:
Step 1: Identify the points
The points are:
- Point A: (1, 5)
- Point B: (4, 1)
- Point C: (−4, −5)
Step 2: Use the Distance Formula
The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is calculated using the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Step 3: Calculate the lengths of the sides
Length of AB: Using points A(1, 5) and B(4, 1): \[ d_{AB} = \sqrt{(4 - 1)^2 + (1 - 5)^2} = \sqrt{(3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
Length of BC: Using points B(4, 1) and C(−4, −5): \[ d_{BC} = \sqrt{(-4 - 4)^2 + (-5 - 1)^2} = \sqrt{(-8)^2 + (-6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \]
Length of CA: Using points C(−4, −5) and A(1, 5): \[ d_{CA} = \sqrt{(1 - (-4))^2 + (5 - (-5))^2} = \sqrt{(1 + 4)^2 + (5 + 5)^2} = \sqrt{(5)^2 + (10)^2} = \sqrt{25 + 100} = \sqrt{125} = 5\sqrt{5} \] Calculating \( 5\sqrt{5} \): \[ \sqrt{5} \approx 2.236 \implies 5\sqrt{5} \approx 5 \times 2.236 \approx 11.18 \]
Step 4: Calculate the perimeter
The perimeter \( P \) of the triangle is the sum of its side lengths: \[ P = d_{AB} + d_{BC} + d_{CA} = 5 + 10 + 5\sqrt{5} \approx 5 + 10 + 11.18 \approx 26.18 \]
Step 5: Round the result
Rounding 26.18 to the nearest tenth gives: \[ \text{Perimeter} \approx 26.2 \]
Final Answer
The perimeter of the triangle, rounded to the nearest tenth, is 26.2 units.
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