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.(4 points)
summarize

Answer 1

How did you get 2.236

Answer 2

To find the perimeter of the triangle formed by the points (1, 5), (4, 1), and (-4, -5), Nathan needs to calculate the lengths of each side of the triangle using the distance formula.

### Step 1: Identify the Points
Label the points as follows:
- Point A = (1, 5)
- Point B = (4, 1)
- Point C = (-4, -5)

### Step 2: Calculate the Lengths of Each Side
Use the distance formula, which is given by:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

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

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

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

### Step 3: Calculate the Perimeter
Add the lengths of all sides to find the perimeter:
\[
\text{Perimeter} = AB + BC + CA = 5 + 10 + 5\sqrt{5}
\]
Using \( \sqrt{5} \approx 2.236 \):
\[
5\sqrt{5} \approx 5 \times 2.236 \approx 11.18
\]
Now, add the lengths together:
\[
\text{Perimeter} \approx 5 + 10 + 11.18 \approx 26.18
\]

### Step 4: Round to the Nearest Tenth
After rounding:
\[
\text{Perimeter} \approx 26.2
\]

### Summary
Nathan calculates the perimeter of the triangle with vertices at (1,5), (4,1), and (-4,-5) to be approximately 26.2 units by summing the lengths of the sides AB (5), BC (10), and CA (approximately 11.2).

Answer 3

To clarify, 2.236 is an approximation of the square root of 5 (\(\sqrt{5}\)). This approximation is approximately 2.236067977 when computed more precisely.

Here's how you can arrive at the value:

### Calculation of \(\sqrt{5}\)

1. **Estimate the value**:
- We know that \(2^2 = 4\) and \(3^2 = 9\), so \(\sqrt{5}\) must be between 2 and 3.

2. **Refine the estimate**:
- A more refined approximation can be done using decimals. We can try 2.2 and 2.3:
- \(2.2^2 = 4.84\)
- \(2.3^2 = 5.29\)
- Since \(4.84 < 5 < 5.29\), we know \(\sqrt{5}\) is between 2.2 and 2.3.

3. **Finding a better approximation**:
- Testing further, you can check:
- \(2.24^2 = 5.0176\) (too high)
- \(2.23^2 = 4.9729\) (too low)
- This indicates that \(\sqrt{5}\) is close to 2.236.

4. **Using a calculator**:
- If you have access to a calculator, you can directly compute \(\sqrt{5}\) to achieve a more precise value of approximately 2.236067977.

### Conclusion
The value of \(\sqrt{5} \approx 2.236\) is commonly used in calculations for convenience and is an accurate enough estimate for most purposes when rounded. Therefore, when calculating \(5\sqrt{5}\), you multiply 5 by this approximation, yielding a value around 11.18, which was then rounded to approximately 11.2 for the perimeter calculation of the triangle.