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 \( A(1, 5) \), \( B(4, 1) \), and \( C(-4, -5) \), we will follow these steps:

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

We will use the distance formula to find the lengths of each side of the triangle. The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:

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

#### Calculate Length of Side \( AB \)

Points \( A(1, 5) \) and \( B(4, 1) \):
\[
AB = \sqrt{(4 - 1)^2 + (1 - 5)^2}
\]
\[
= \sqrt{(3)^2 + (-4)^2}
\]
\[
= \sqrt{9 + 16}
\]
\[
= \sqrt{25} = 5
\]

#### Calculate Length of Side \( BC \)

Points \( B(4, 1) \) and \( C(-4, -5) \):
\[
BC = \sqrt{(-4 - 4)^2 + (-5 - 1)^2}
\]
\[
= \sqrt{(-8)^2 + (-6)^2}
\]
\[
= \sqrt{64 + 36}
\]
\[
= \sqrt{100} = 10
\]

#### Calculate Length of Side \( CA \)

Points \( C(-4, -5) \) and \( A(1, 5) \):
\[
CA = \sqrt{(1 - (-4))^2 + (5 - (-5))^2}
\]
\[
= \sqrt{(1 + 4)^2 + (5 + 5)^2}
\]
\[
= \sqrt{(5)^2 + (10)^2}
\]
\[
= \sqrt{25 + 100}
\]
\[
= \sqrt{125} = 5\sqrt{5} \approx 11.2
\]

### Step 2: Calculate the Perimeter

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

\[
P = AB + BC + CA
\]
\[
= 5 + 10 + 5\sqrt{5}
\]
Using \( \sqrt{5} \approx 2.236 \):

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

Now we can approximate the perimeter:

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

### Final Step: Round to the Nearest Tenth

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

\[
P \approx 26.2
\]

### Conclusion

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