To find the perimeter of the triangle formed by the points \(A(1,5)\), \(B(4,1)\), and \(C(-4,-5)\) on the coordinate grid, Nathan needs to follow these steps:
Step 1: Calculate the Length of Each Side of the Triangle
The perimeter of a triangle is the sum of the lengths of its sides. We will calculate the lengths of sides \(AB\), \(BC\), and \(CA\) using the distance formula, which is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Length of Side AB
For 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 Side BC
For points \(B(4, 1)\) and \(C(-4, -5)\):
\[ d_{BC} = \sqrt{(-4 - 4)^2 + (-5 - 1)^2} \] \[ = \sqrt{(-8)^2 + (-6)^2} \] \[ = \sqrt{64 + 36} = \sqrt{100} = 10 \]
Length of Side CA
For points \(C(-4, -5)\) and \(A(1, 5)\):
\[ d_{CA} = \sqrt{(1 - (-4))^2 + (5 - (-5))^2} \] \[ = \sqrt{(1 + 4)^2 + (5 + 5)^2} \] \[ = \sqrt{(5)^2 + (10)^2} \] \[ = \sqrt{25 + 100} = \sqrt{125} = 5\sqrt{5} \approx 11.2 \]
Step 2: Sum the Lengths to Find the Perimeter
Now that we have the lengths of all three sides:
- \(AB = 5\)
- \(BC = 10\)
- \(CA \approx 11.2\)
The perimeter \(P\) of the triangle is calculated as follows:
\[ P = AB + BC + CA \] \[ P = 5 + 10 + 11.2 \] \[ P = 26.2 \]
Step 3: Round the Result to the Nearest Tenth
The perimeter calculated is \(26.2\), and already it is rounded to the nearest tenth.
Conclusion
The perimeter of Nathan's triangle is approximately 26.2 units.
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