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..

Answer 1

To find the perimeter of the triangle formed by the points \( A(1, 5) \), \( B(4, 1) \), and \( C(-4, -5) \), we need to calculate the lengths of the sides of the triangle and then sum those lengths.

### Step 1: Calculate the lengths of each side

1. **Length of side AB**:
The distance \( d \) between points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) can be calculated using the distance formula:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
For the points \( A(1, 5) \) and \( B(4, 1) \):
\[
d_{AB} = \sqrt{(4 - 1)^2 + (1 - 5)^2} = \sqrt{(3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\]

2. **Length of side BC**:
For points \( B(4, 1) \) and \( C(-4, -5) \):
\[
d_{BC} = \sqrt{(-4 - 4)^2 + (-5 - 1)^2} = \sqrt{(-8)^2 + (-6)^2} = \sqrt{64 + 36} = \sqrt{100} = 10
\]

3. **Length of side CA**:
For points \( C(-4, -5) \) and \( A(1, 5) \):
\[
d_{CA} = \sqrt{(1 - (-4))^2 + (5 - (-5))^2} = \sqrt{(1 + 4)^2 + (5 + 5)^2} = \sqrt{5^2 + 10^2} = \sqrt{25 + 100} = \sqrt{125} = 5\sqrt{5}
\]

### Step 2: Calculate the perimeter
Now, we can find the perimeter \( P \) of the triangle by adding the lengths of the three sides:
\[
P = d_{AB} + d_{BC} + d_{CA} = 5 + 10 + 5\sqrt{5}
\]

### Step 3: Approximate the perimeter
To find the numerical value of the perimeter, we'll evaluate \( 5\sqrt{5} \) and add it to 15.

First, calculate \( \sqrt{5} \):
\[
\sqrt{5} \approx 2.236
\]

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

Now add this to the other lengths:
\[
P \approx 5 + 10 + 11.18 = 26.18
\]

### Step 4: Round to the nearest tenth
Finally, rounding \( 26.18 \) to the nearest tenth gives us:
\[
\text{Perimeter} \approx 26.2
\]

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