How Do You Find the Perimeter of a Triangle with Vertices (1, 1), (4, 1), and (4, 5)?

Answer – To find the perimeter of a triangle with coordinates/vertices given, use the distance formula $\sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}$ . The perimeter of the given triangle is found to be 12 units.

Explanation:

Let’s begin by graphing the given triangle and labeling its vertices.

A triangle with vertices A, B, and C

Now, since the perimeter of a triangle is the sum of its sides:

$T h e p e r i m e t e r o f t h e t r i a n g l e A B C = A B + B C + C A$

To find each of these sides, we must use the distance formula, which allows us to find the distance between any two points (x₁, y₁) and (x₂, y₂) on a coordinate plane. It is given by:

$\sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}$

For the side AB, (x₁, y₁) = (1, 1) and (x₂, y₂) = (4, 1).

Now, using the distance formula:

$A B = \sqrt{{(4 - 1)}^{2} + {(1 - 1)}^{2}}$

$= \sqrt{3^{2} + 0}$

$= \sqrt{9}$

$= 3$

$A B = 3$ units

Similarly, for side BC, (x₁, y₁) = (4, 1) and (x₂, y₂) = (4, 5).

$B C = \sqrt{{(4 - 4)}^{2} + {(5 - 1)}^{2}}$

$= \sqrt{0 + 4^{2}}$

$= \sqrt{16}$

$= 4$

$B C = 4$ units

Finally, for side CA, (x₁, y₁) = (4, 5) and (x₂, y₂) = (1, 1).

$C A = \sqrt{{(1 - 4)}^{2} + {(1 - 5)}^{2}}$

$= \sqrt{{(- 3)}^{2} + {(- 4)}^{2}}$

$= \sqrt{9 + 16}$

$= \sqrt{25}$

$= 5$

$C A = 5$ units

Since $t h e p e r i m e t e r o f t h e t r i a n g l e A B C = A B + B C + C A$ :

$P e r i m e t e r = 3 + 4 + 5 = 12$

Thus, the perimeter of the given triangle is 12 units.

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Answer – To find the perimeter of a triangle with coordinates/vertices given, use the distance formula (x2 – x1)2 + (y2 – y1)2 . The perimeter of the given triangle is found to be 12 units.

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Answer – To find the perimeter of a triangle with coordinates/vertices given, use the distance formula $\sqrt{{(x_{2} - x_{1})}^{2} + {(y_{2} - y_{1})}^{2}}$ . The perimeter of the given triangle is found to be 12 units.