Question

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? Show all steps that Nathan must take to find the solution.

Answer 1

simplified

Answer 2

To find the perimeter of the triangle formed by the points \((1, 5)\), \((4, 1)\), and \((-4, -5)\), we need to determine the lengths of the sides of the triangle. The perimeter is the sum of these lengths.

### Step 1: Determine the lengths of the sides
- Let \(A(1, 5)\), \(B(4, 1)\), and \(C(-4, -5)\). We will calculate the distances \(AB\), \(BC\), and \(CA\).

**Length of AB:**
Using the distance formula:
\[
AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
where \(A(1, 5)\) and \(B(4, 1)\):
\[
AB = \sqrt{(4 - 1)^2 + (1 - 5)^2} = \sqrt{(3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\]

**Length of BC:**
Using the distance formula where \(B(4, 1)\) and \(C(-4, -5)\):
\[
BC = \sqrt{(-4 - 4)^2 + (-5 - 1)^2} = \sqrt{(-8)^2 + (-6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10
\]

**Length of CA:**
Using the distance formula where \(C(-4, -5)\) and \(A(1, 5)\):
\[
CA = \sqrt{(1 - (-4))^2 + (5 - (-5))^2} = \sqrt{(1 + 4)^2 + (5 + 5)^2} = \sqrt{(5)^2 + (10)^2} = \sqrt{25 + 100} = \sqrt{125} = 5\sqrt{5}
\]
Calculating \(5\sqrt{5}\):
\[
\sqrt{5} \approx 2.236 \implies 5\sqrt{5} \approx 5 \times 2.236 = 11.18
\]

### Step 2: Calculate the perimeter
Now, we add the lengths of the sides:
\[
\text{Perimeter} = AB + BC + CA = 5 + 10 + 5\sqrt{5}
\]
To get the numerical value of the perimeter:
\[
\text{Perimeter} \approx 5 + 10 + 11.18 = 26.18
\]

### Step 3: Round to the nearest tenth
Rounding \(26.18\) to the nearest tenth, we get:
\[
26.2
\]

Thus, the perimeter of the triangle, rounded to the nearest tenth of a unit, is \(\boxed{26.2}\).