Question

ab1bc for a(-3,2) and c(2,7). which of the following could be the coordinates of b? select all that apply. a. (-2,2) b. (-3,7) c. (-4,5) d. (1,3) e. (-1,-1) f. (8,0)

Explanation:

Step1: Recall triangle - vertex property

For three points \(A(x_1,y_1)\), \(B(x_2,y_2)\) and \(C(x_3,y_3)\) to form a triangle, the three points should not be collinear.

Step2: Use the slope - formula

The slope between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let \(A(-3,2)\) and \(C(2,7)\). The slope of \(AC\) is \(m_{AC}=\frac{7 - 2}{2+3}=\frac{5}{5}=1\).

Step3: Calculate slopes for each option

For option A \((-2,2)\):
The slope of \(AB\) with \(A(-3,2)\) and \(B(-2,2)\) is \(m_{AB}=\frac{2 - 2}{-2 + 3}=0
eq1\), so it can be a vertex of the triangle.
For option B \((-3,7)\):
The slope of \(AB\) with \(A(-3,2)\) and \(B(-3,7)\) is undefined (since \(x_1=x_2=-3\)), and it is not collinear with \(AC\), so it can be a vertex of the triangle.
For option C \((-4,5)\):
The slope of \(AB\) with \(A(-3,2)\) and \(B(-4,5)\) is \(m_{AB}=\frac{5 - 2}{-4+3}=-3
eq1\), so it can be a vertex of the triangle.
For option D \((1,3)\):
The slope of \(AB\) with \(A(-3,2)\) and \(B(1,3)\) is \(m_{AB}=\frac{3 - 2}{1 + 3}=\frac{1}{4}
eq1\), so it can be a vertex of the triangle.
For option E \((-1,-1)\):
The slope of \(AB\) with \(A(-3,2)\) and \(B(-1,-1)\) is \(m_{AB}=\frac{-1 - 2}{-1+3}=-\frac{3}{2}
eq1\), so it can be a vertex of the triangle.
For option F \((8,0)\):
The slope of \(AB\) with \(A(-3,2)\) and \(B(8,0)\) is \(m_{AB}=\frac{0 - 2}{8 + 3}=-\frac{2}{11}
eq1\), so it can be a vertex of the triangle.

Answer:

A. \((-2,2)\), B. \((-3,7)\), C. \((-4,5)\), D. \((1,3)\), E. \((-1,-1)\), F. \((8,0)\)