The perimeter of the triangle whose vertices are (- 1,4), ( - 4, - 2), (3, - 4), will be

Options

A. 38
B. 16
C. 42
D. None of these

Answer

To find the perimeter of the triangle with vertices at A(-1, 4), B(-4, -2), and C(3, -4), we will follow these steps:

### Step 1: Calculate the length of side AB
Using the distance formula:
\[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of points A and B:
\[ AB = \sqrt{((-4) - (-1))^2 + ((-2) - 4)^2} \]
\[ = \sqrt{(-4 + 1)^2 + (-2 - 4)^2} \]
\[ = \sqrt{(-3)^2 + (-6)^2} \]
\[ = \sqrt{9 + 36} = \sqrt{45} \]

### Step 2: Calculate the length of side BC
Using the distance formula again:
\[ BC = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of points B and C:
\[ BC = \sqrt{(3 - (-4))^2 + ((-4) - (-2))^2} \]
\[ = \sqrt{(3 + 4)^2 + (-4 + 2)^2} \]
\[ = \sqrt{(7)^2 + (-2)^2} \]
\[ = \sqrt{49 + 4} = \sqrt{53} \]

### Step 3: Calculate the length of side CA
Using the distance formula:
\[ CA = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of points C and A:
\[ CA = \sqrt{((-1) - 3)^2 + (4 - (-4))^2} \]
\[ = \sqrt{(-4)^2 + (4 + 4)^2} \]
\[ = \sqrt{16 + 64} = \sqrt{80} \]

### Step 4: Calculate the perimeter of the triangle
The perimeter \( P \) is the sum of the lengths of all sides:
\[ P = AB + BC + CA \]
Substituting the lengths we calculated:
\[ P = \sqrt{45} + \sqrt{53} + \sqrt{80} \]

### Step 5: Approximate the values
Calculating the approximate values:
\[ \sqrt{45} \approx 6.71, \quad \sqrt{53} \approx 7.28, \quad \sqrt{80} \approx 8.94 \]
Adding these:
\[ P \approx 6.71 + 7.28 + 8.94 \approx 22.93 \]

Thus, the perimeter of the triangle is approximately 22.93.

### Final Answer
The perimeter of the triangle is approximately \( 22.93 \).

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