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.

Answer 1

To find the perimeter of the triangle formed by the points \((1, 5)\), \((4, 1)\), and \((-4, -5)\), we need to calculate the lengths of each side using the distance formula, which is given by:

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

Let's denote the points as follows:
- \(A(1, 5)\)
- \(B(4, 1)\)
- \(C(-4, -5)\)

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

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

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

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

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

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

### Step 4: Calculate the perimeter

Now, we can find the perimeter by adding the lengths of the sides:

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

Since \(5\sqrt{5} \approx 11.2\), we can evaluate:

\[
\text{Perimeter} \approx 5 + 10 + 11.2 = 26.2
\]

### Step 5: Round to the nearest tenth

Thus, the perimeter of the triangle, rounded to the nearest tenth of a unit, is:

\[
\boxed{26.2}
\]