Question

a triangle has the given coordinates: a(-1,0), b(2,-3), and c(4,1). what is the equation of the median line ad for the triangle? a. y = 1/4x + 1/4 b. y = -1/2x + 1/2 c. y = 1/2x - 1/2 d. y = -1/4x - 1/4

Explanation:

Step1: Find the mid - point of \(BC\)

The mid - point formula for two points \((x_1,y_1)\) and \((x_2,y_2)\) is \((\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\).
For \(B(2,-3)\) and \(C(4,1)\), \(x=\frac{2 + 4}{2}=3\), \(y=\frac{-3+1}{2}=-1\). So the mid - point \(D\) is \((3,-1)\).

Step2: Find the slope of \(AD\)

The slope formula for two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(m=\frac{y_2 - y_1}{x_2 - x_1}\).
For \(A(-1,0)\) and \(D(3,-1)\), \(m=\frac{-1 - 0}{3+1}=-\frac{1}{4}\).

Step3: Use the point - slope form \(y - y_1=m(x - x_1)\)

Using point \(A(-1,0)\) and \(m =-\frac{1}{4}\), \(y-0=-\frac{1}{4}(x + 1)\), which simplifies to \(y=-\frac{1}{4}x-\frac{1}{4}\).

Answer:

D. \(y =-\frac{1}{4}x-\frac{1}{4}\)