A triangle is graphed on a standard coordindate plane. What is the perimeter of the triangle if it has vertices `(1, 4), (1, 7), and (4, 4)`?

Options

A. `3+sqrt(2)`
B. `3sqrt(2)`
C. `6+3sqrt(2)`
D. `9+sqrt(2)`

Answer

To find the perimeter of the triangle with vertices at (1, 4), (1, 7), and (4, 4), we will follow these steps:

### Step 1: Identify the vertices
The vertices of the triangle are:
- A(1, 4)
- B(1, 7)
- C(4, 4)

### Step 2: Calculate the lengths of the sides
We will use the distance formula to find the lengths of the sides of the triangle. The distance formula between two points (x1, y1) and (x2, y2) is given by:
\[ d = \sqrt{(x2 - x1)^2 + (y2 - y1)^2} \]

#### Finding AB:
Using points A(1, 4) and B(1, 7):
\[ AB = \sqrt{(1 - 1)^2 + (7 - 4)^2} \]
\[ AB = \sqrt{0 + (3)^2} \]
\[ AB = \sqrt{9} \]
\[ AB = 3 \]

#### Finding BC:
Using points B(1, 7) and C(4, 4):
\[ BC = \sqrt{(4 - 1)^2 + (4 - 7)^2} \]
\[ BC = \sqrt{(3)^2 + (-3)^2} \]
\[ BC = \sqrt{9 + 9} \]
\[ BC = \sqrt{18} \]
\[ BC = 3\sqrt{2} \]

#### Finding AC:
Using points A(1, 4) and C(4, 4):
\[ AC = \sqrt{(4 - 1)^2 + (4 - 4)^2} \]
\[ AC = \sqrt{(3)^2 + 0} \]
\[ AC = \sqrt{9} \]
\[ AC = 3 \]

### Step 3: Calculate the perimeter
The perimeter P of the triangle is the sum of the lengths of its sides:
\[ P = AB + BC + AC \]
\[ P = 3 + 3\sqrt{2} + 3 \]
\[ P = 6 + 3\sqrt{2} \]

### Final Answer:
The perimeter of the triangle is \( 6 + 3\sqrt{2} \).

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