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

To find the perimeter of the triangle formed by the points \(A(1,5)\), \(B(4,1)\), and \(C(-4,-5)\) on the coordinate grid, Nathan needs to follow these steps:

### Step 1: Calculate the Length of Each Side of the Triangle

The perimeter of a triangle is the sum of the lengths of its sides. We will calculate the lengths of sides \(AB\), \(BC\), and \(CA\) using the distance formula, which is given by:

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

#### Length of Side AB

For points \(A(1, 5)\) and \(B(4, 1)\):

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

#### Length of Side BC

For points \(B(4, 1)\) and \(C(-4, -5)\):

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

#### Length of Side CA

For points \(C(-4, -5)\) and \(A(1, 5)\):

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

### Step 2: Sum the Lengths to Find the Perimeter

Now that we have the lengths of all three sides:

- \(AB = 5\)
- \(BC = 10\)
- \(CA \approx 11.2\)

The perimeter \(P\) of the triangle is calculated as follows:

\[
P = AB + BC + CA
\]
\[
P = 5 + 10 + 11.2
\]
\[
P = 26.2
\]

### Step 3: Round the Result to the Nearest Tenth

The perimeter calculated is \(26.2\), and already it is rounded to the nearest tenth.

### Conclusion

The perimeter of Nathan's triangle is approximately **26.2 units**.