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

To find the perimeter of the triangle formed by the points \((1, 5)\), \((4, 1)\), and \((-4, -5)\), we'll first find the lengths of each side of the triangle using the distance formula. The distance \(d\) 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 denote the points as follows:
- \(A(1, 5)\)
- \(B(4, 1)\)
- \(C(-4, -5)\)

We'll calculate the lengths of sides \(AB\), \(BC\), and \(CA\).

### Step 1: Calculate the length of side \(AB\)

Using the distance formula:
\[
d_{AB} = \sqrt{(4 - 1)^2 + (1 - 5)^2}
\]
Calculating the differences:
\[
= \sqrt{(3)^2 + (-4)^2}
\]
\[
= \sqrt{9 + 16}
\]
\[
= \sqrt{25}
\]
\[
= 5
\]

### Step 2: Calculate the length of side \(BC\)

Now for side \(BC\):
\[
d_{BC} = \sqrt{(-4 - 4)^2 + (-5 - 1)^2}
\]
Calculating the differences:
\[
= \sqrt{(-8)^2 + (-6)^2}
\]
\[
= \sqrt{64 + 36}
\]
\[
= \sqrt{100}
\]
\[
= 10
\]

### Step 3: Calculate the length of side \(CA\)

Finally for side \(CA\):
\[
d_{CA} = \sqrt{(1 - (-4))^2 + (5 - (-5))^2}
\]
Calculating the differences:
\[
= \sqrt{(1 + 4)^2 + (5 + 5)^2}
\]
\[
= \sqrt{(5)^2 + (10)^2}
\]
\[
= \sqrt{25 + 100}
\]
\[
= \sqrt{125}
\]
\[
= 5\sqrt{5} \approx 11.18 \quad (\text{Using } \sqrt{5} \approx 2.236)
\]

### Step 4: Calculate the perimeter

Now, we can add these lengths together to find the perimeter \(P\):

\[
P = d_{AB} + d_{BC} + d_{CA}
\]
\[
= 5 + 10 + 5\sqrt{5}
\]
\[
\approx 5 + 10 + 11.18 \approx 26.18
\]

### Step 5: Round to the nearest tenth

Rounding \(26.18\) to the nearest tenth gives us:

\[
\boxed{26.2}
\]

Thus, the perimeter of the triangle is approximately \(26.2\) units.