Question

1. the triangle below is graphed on the coordinate plane. two of its vertices and t midpoint of the longer leg are labeled. find the perimeter of the triangle. (3 pts, pts work, 1 pt answer)

(-5, 3)
(-5, 1) (-2, 1)
perimeter =
type a response

Explanation:

Step1: Find length of horizontal leg

The horizontal leg is between \((-5, 1)\) and \((-2, 1)\). Since the y - coordinates are the same, the length is the absolute difference of x - coordinates.
Length \(d_1=\vert - 2-(-5)\vert=\vert - 2 + 5\vert = 3\)

Step2: Find length of vertical leg

The mid - point of the longer leg is \((-5,3)\) and one end of the vertical leg is \((-5,1)\). Let the other end of the vertical leg be \((-5,y)\). The mid - point formula for two points \((x_1,y_1)\) and \((x_2,y_2)\) is \((\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\). Here \(x_1=x_2=-5\), \(\frac{y_1 + y_2}{2}=3\) and \(y_1 = 1\). So \(\frac{1 + y_2}{2}=3\), \(1 + y_2=6\), \(y_2 = 5\). The vertical leg is between \((-5,1)\) and \((-5,5)\). Since x - coordinates are the same, length \(d_2=\vert5 - 1\vert=4\)

Step3: Find length of hypotenuse

Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) for points \((-5,5)\) and \((-2,1)\). \(x_1=-5,y_1 = 5,x_2=-2,y_2 = 1\). Then \(d_3=\sqrt{(-2+5)^2+(1 - 5)^2}=\sqrt{3^2+(-4)^2}=\sqrt{9 + 16}=\sqrt{25}=5\)

Step4: Calculate perimeter

Perimeter \(P=d_1 + d_2 + d_3=3 + 4+5 = 12\)

Answer:

12