Hello!
To find the perimeter of the triangle, we need to find the length of all the sides using the distance formula.
The distance formula is: [tex] d =\sqrt{(x_2-x_1)^2+(y_2-y_1)^2} [/tex].
First, we can find the distance between the points (-3, -1) and (2, -1). The point (-3, -1) can be assigned to [tex] (x_{1},y_{1}) [/tex], and (2, -1) is assigned to [tex] (x_{2},y_{2}) [/tex]. Then, substitute the values into the formula.
[tex] d =\sqrt{(2 - (-3))^{2}+ (-1 - (-1))^{2}} [/tex]
[tex] d =\sqrt{5^{2}+0^{2}} [/tex]
[tex] d =\sqrt{25} = 5 [/tex]
The distance between the points (-3, -1) and (2, -1) is 5 units.
Secondly, we need to find the distance between the points (2, 3) and (2, -1). Assign those points to [tex] (x_{1},y_{1}) [/tex] and [tex] (x_{2},y_{2}) [/tex], then substitute it into the formula.
[tex] d =\sqrt{(2 - 2)^{2}+ (-1 - 3)^{2}} [/tex]
[tex] d =\sqrt{0^{2}+(-4)^{2}} [/tex]
[tex] d =\sqrt{16} = 4 [/tex]
The distance between the two points (2, 3) and (2, -1) is 4 units.
Finally, we use the distance formula again to find the distance between the points (-3, -1) and (2, 3). Remember the assign the ordered pairs to [tex] (x_{1},y_{1}) [/tex] and [tex] (x_{2},y_{2}) [/tex] and substitute!
[tex] d =\sqrt{(2 -(-3))^{2}+ (3 - (-1))^{2}} [/tex]
[tex] d =\sqrt{5^{2}+4^{2}} [/tex]
[tex] d =\sqrt{25 + 16} [/tex]
[tex] d =\sqrt{41} [/tex] This is equal to approximately 6.40 units.
The last step is to find the perimeter. To find the perimeter, add of the three sides of the triangle together.
P = 5 units + 4 units + 6.4 units
P = 15.4 units
Therefore, the perimeter of this triangle is choice A, 15.4.
