Nathan draws a triangle on the coordinate grid. He marks his points at (1,5) , (4,1) , and (−4,−5) . What is the perimeter of the triangle, rounded to the nearest tenth of a unit? Explain the steps that Nathan must take to find the solution.

Answer 1

To find the perimeter of the triangle with vertices at the points \( A(1,5) \), \( B(4,1) \), and \( C(-4,-5) \), Nathan should follow these steps:

1. Calculate the lengths of the sides of the triangle \( AB \), \( BC \), and \( CA \) using the distance formula. The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

2. Calculate the length of side \( AB \):

   • Coordinates of \( A \) are \( (1,5) \) and \( B \) are \( (4,1) \): \[ AB = \sqrt{(4 - 1)^2 + (1 - 5)^2} \] \[ = \sqrt{(3)^2 + (-4)^2} \] \[ = \sqrt{9 + 16} = \sqrt{25} = 5 \]

3. Calculate the length of side \( BC \):

   • Coordinates of \( B \) are \( (4,1) \) and \( C \) are \( (-4,-5) \): \[ BC = \sqrt{((-4) - 4)^2 + ((-5) - 1)^2} \] \[ = \sqrt{(-8)^2 + (-6)^2} \] \[ = \sqrt{64 + 36} = \sqrt{100} = 10 \]

4. Calculate the length of side \( CA \):

   • Coordinates of \( C \) are \( (-4,-5) \) and \( A \) are \( (1,5) \): \[ CA = \sqrt{(1 - (-4))^2 + (5 - (-5))^2} \] \[ = \sqrt{(1 + 4)^2 + (5 + 5)^2} \] \[ = \sqrt{(5)^2 + (10)^2} \] \[ = \sqrt{25 + 100} = \sqrt{125} = 5\sqrt{5} \approx 11.1803 \]

5. Add the lengths of the sides to find the perimeter: \[ \text{Perimeter} = AB + BC + CA \] \[ = 5 + 10 + 5\sqrt{5} \approx 5 + 10 + 11.1803 = 26.1803 \]

6. Round the perimeter to the nearest tenth: \[ \approx 26.2 \]

In summary, Nathan finds the lengths of each side using the distance formula, adds the lengths to calculate the perimeter, and rounds the result to the nearest tenth. The final answer is that the perimeter of the triangle is approximately 26.2 units.

Answer 2

make it all regular text

Answer 3

To find the perimeter of the triangle with vertices at the points A(1,5), B(4,1), and C(-4,-5), Nathan should follow these steps:

1. Calculate the lengths of the sides of the triangle AB, BC, and CA using the distance formula. The distance \( d \) between two points (x1, y1) and (x2, y2) is given by the formula:

   \( d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2} \).

2. Calculate the length of side AB:

   • Coordinates of A are (1,5) and B are (4,1):
   • \( AB = \sqrt{(4 - 1)^2 + (1 - 5)^2} \).
   • \( = \sqrt{3^2 + (-4)^2} \).
   • \( = \sqrt{9 + 16} = \sqrt{25} = 5 \).

3. Calculate the length of side BC:

   • Coordinates of B are (4,1) and C are (-4,-5):
   • \( BC = \sqrt{((-4) - 4)^2 + ((-5) - 1)^2} \).
   • \( = \sqrt{(-8)^2 + (-6)^2} \).
   • \( = \sqrt{64 + 36} = \sqrt{100} = 10 \).

4. Calculate the length of side CA:

   • Coordinates of C are (-4,-5) and A are (1,5):
   • \( CA = \sqrt{(1 - (-4))^2 + (5 - (-5))^2} \).
   • \( = \sqrt{(1 + 4)^2 + (5 + 5)^2} \).
   • \( = \sqrt{5^2 + 10^2} \).
   • \( = \sqrt{25 + 100} = \sqrt{125} = 5\sqrt{5} \approx 11.1803 \).

5. Add the lengths of the sides to find the perimeter:

   • \( \text{Perimeter} = AB + BC + CA \).
   • \( = 5 + 10 + 5\sqrt{5} \).
   • \( \approx 5 + 10 + 11.1803 = 26.1803 \).

6. Round the perimeter to the nearest tenth:

   • \( \approx 26.2 \).

In summary, Nathan finds the lengths of each side using the distance formula, adds the lengths to calculate the perimeter, and rounds the result to the nearest tenth. The final answer is that the perimeter of the triangle is approximately 26.2 units.

Answer 4

equations and stuff too

make that regular text

Answer 5

To find the perimeter of the triangle with vertices at the points A(1,5), B(4,1), and C(-4,-5), Nathan should follow these steps:

1. Calculate the lengths of the sides of the triangle AB, BC, and CA using the distance formula. The distance \( d \) between two points (x1, y1) and (x2, y2) is given by the formula:

   d = √((x2 - x1)² + (y2 - y1)²).

2. For the length of side AB:

   • The coordinates of A are (1,5) and B are (4,1).
   • Using the distance formula: AB = √((4 - 1)² + (1 - 5)²) = √(3² + (-4)²) = √(9 + 16) = √25 = 5.

3. For the length of side BC:

   • The coordinates of B are (4,1) and C are (-4,-5).
   • Using the distance formula: BC = √((-4 - 4)² + (-5 - 1)²) = √((-8)² + (-6)²) = √(64 + 36) = √100 = 10.

4. For the length of side CA:

   • The coordinates of C are (-4,-5) and A are (1,5).
   • Using the distance formula: CA = √((1 - (-4))² + (5 - (-5))²) = √((1 + 4)² + (5 + 5)²) = √(5² + 10²) = √(25 + 100) = √125 = 5√5 ≈ 11.1803.

5. Add the lengths of the sides to find the perimeter:

   • The perimeter is given by: Perimeter = AB + BC + CA = 5 + 10 + 5√5 ≈ 5 + 10 + 11.1803 ≈ 26.1803.

6. Round the perimeter to the nearest tenth:

   • The rounded perimeter is approximately 26.2.

In summary, Nathan calculates the lengths of each side using the distance formula, adds these lengths to determine the perimeter, and rounds the result to the nearest tenth. The final answer is that the perimeter of the triangle is approximately 26.2 units.