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? What is the area also?

Answer 1

To find the perimeter and area of the triangle with the vertices at the points \( A(1, 5) \), \( B(4, 1) \), and \( C(-4, -5) \), we need to calculate the lengths of the sides of the triangle first.

### Step 1: Calculate the lengths of the sides.

#### Length of side \( AB \):
Using the distance formula:

\[
AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
Where \( A(1, 5) \) and \( B(4, 1) \):

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

#### Length of side \( BC \):
Using the distance formula for points \( B(4, 1) \) and \( C(-4, -5) \):

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

#### Length of side \( CA \):
Using the distance formula for 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:

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

Calculating \( 5\sqrt{5} \):

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

Thus,

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

Rounding to the nearest tenth, the perimeter is:

\[
P \approx 26.2
\]

### Step 3: Calculate the area.
We can calculate the area \( A \) of the triangle using the formula:

\[
A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|
\]

Substituting in the coordinates of points \( A(1, 5) \), \( B(4, 1) \), \( C(-4, -5) \):

- \( x_1 = 1, y_1 = 5 \)
- \( x_2 = 4, y_2 = 1 \)
- \( x_3 = -4, y_3 = -5 \)

Calculating:

\[
A = \frac{1}{2} \left| 1(1 - (-5)) + 4((-5) - 5) + (-4)(5 - 1) \right|
\]

Now, calculate each term:

1. \( 1(1 + 5) = 1 \times 6 = 6 \)
2. \( 4(-10) = -40 \)
3. \( -4(4) = -16 \)

Putting it all together:

\[
A = \frac{1}{2} \left| 6 - 40 - 16 \right| = \frac{1}{2} \left| 6 - 56 \right| = \frac{1}{2} \left| -50 \right| = \frac{1}{2} \times 50 = 25
\]

### Conclusion:
The perimeter of the triangle is:

\[
\boxed{26.2}
\]

The area of the triangle is:

\[
\boxed{25}
\]