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? Explain the steps that Nathan must take to find the solution.(4 points)

Answer 1

make it all regular text

Answer 2

To find the perimeter of the triangle formed by the points (1, 5), (4, 1), and (-4, -5), Nathan can follow these steps:

### Step 1: Calculate the distances between each pair of points using the distance formula.

The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Let’s label the points as follows:
- Point A: \((1, 5)\)
- Point B: \((4, 1)\)
- Point C: \((-4, -5)\)

### Step 2: Calculate the distance AB

Using the coordinates of points A and B:
- \(x_1 = 1\), \(y_1 = 5\)
- \(x_2 = 4\), \(y_2 = 1\)

\[
AB = \sqrt{(4 - 1)^2 + (1 - 5)^2} = \sqrt{(3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\]

### Step 3: Calculate the distance BC

Using the coordinates of points B and C:
- \(x_1 = 4\), \(y_1 = 1\)
- \(x_2 = -4\), \(y_2 = -5\)

\[
BC = \sqrt{(-4 - 4)^2 + (-5 - 1)^2} = \sqrt{(-8)^2 + (-6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10
\]

### Step 4: Calculate the distance CA

Using the coordinates of points C and A:
- \(x_1 = -4\), \(y_1 = -5\)
- \(x_2 = 1\), \(y_2 = 5\)

\[
CA = \sqrt{(1 - (-4))^2 + (5 - (-5))^2} = \sqrt{(1 + 4)^2 + (5 + 5)^2} = \sqrt{(5)^2 + (10)^2} = \sqrt{25 + 100} = \sqrt{125} = 5\sqrt{5} \approx 11.2
\]

### Step 5: Calculate the perimeter of the triangle

The perimeter \(P\) is the sum of the lengths of all sides:

\[
P = AB + BC + CA = 5 + 10 + 5\sqrt{5}
\]

Calculating \(5\sqrt{5}\) using the approximate value of \(\sqrt{5} \approx 2.236\):

\[
5\sqrt{5} \approx 5 \times 2.236 \approx 11.18
\]

Now summing all distances:

\[
P \approx 5 + 10 + 11.18 = 26.18
\]

### Step 6: Round the perimeter to the nearest tenth

The perimeter \(P \approx 26.18\) rounds to \(26.2\).

### Final Answer

The perimeter of the triangle, rounded to the nearest tenth of a unit, is **26.2**.