Question

Nathan draws a triangle on the coordinate grid. He marks his poinjavascript:void(0)ts at (1,5) ( 1 , 5 ) , (4,1) ( 4 , 1 ) , and (−4,−5) ( − 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.(4 points) simple and basic

Answer 1

To find the perimeter of the triangle defined by the points \( A(1, 5) \), \( B(4, 1) \), and \( C(-4, -5) \), Nathan needs to calculate the lengths of the sides of the triangle (AB, BC, and CA) and then sum these lengths. Here are the steps:

### Step 1: Calculate the length of each side

#### 1.1 Calculate the length of AB

The distance formula is given by:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
For points \( A(1, 5) \) and \( B(4, 1) \):
- \( x_1 = 1 \), \( y_1 = 5 \)
- \( x_2 = 4 \), \( y_2 = 1 \)

Now, substituting the values into the formula:
\[
AB = \sqrt{(4 - 1)^2 + (1 - 5)^2}
\]
\[
= \sqrt{(3)^2 + (-4)^2}
\]
\[
= \sqrt{9 + 16}
\]
\[
= \sqrt{25}
\]
\[
= 5
\]

#### 1.2 Calculate the length of BC

Now for points \( B(4, 1) \) and \( C(-4, -5) \):
- \( x_1 = 4 \), \( y_1 = 1 \)
- \( x_2 = -4 \), \( y_2 = -5 \)

Substituting into the distance formula:
\[
BC = \sqrt{(-4 - 4)^2 + (-5 - 1)^2}
\]
\[
= \sqrt{(-8)^2 + (-6)^2}
\]
\[
= \sqrt{64 + 36}
\]
\[
= \sqrt{100}
\]
\[
= 10
\]

#### 1.3 Calculate the length of CA

Now for points \( C(-4, -5) \) and \( A(1, 5) \):
- \( x_1 = -4 \), \( y_1 = -5 \)
- \( x_2 = 1 \), \( y_2 = 5 \)

Substituting into the distance formula:
\[
CA = \sqrt{(1 - (-4))^2 + (5 - (-5))^2}
\]
\[
= \sqrt{(1 + 4)^2 + (5 + 5)^2}
\]
\[
= \sqrt{(5)^2 + (10)^2}
\]
\[
= \sqrt{25 + 100}
\]
\[
= \sqrt{125}
\]
\[
= 5\sqrt{5} \approx 11.2 \quad (\text{since } \sqrt{5} \approx 2.236)
\]

### Step 2: Calculate the perimeter

Now the perimeter \( P \) of triangle \( ABC \) can be found by summing the lengths of the sides:
\[
P = AB + BC + CA
\]
\[
P = 5 + 10 + 11.2
\]
\[
= 26.2
\]

### Step 3: Round to the nearest tenth

The perimeter of the triangle, rounded to the nearest tenth, is:
\[
\boxed{26.2}
\]