Question

annie is creating a stencil for her artwork using a coordinate plane. the beginning of the left edge of the stencil falls at (-1, 2). she wants to align an important detail on the left edge of her stencil at (2, 1). she knows this is 1:2 of the way to where she wants the end of the stencil. where is the end of the stencil located? (4 points) (0, 5) (8, -1) (0.5, 1.5) (1, 1.7)

Explanation:

Step1: Recall the section formula

The section formula for a point \((x,y)\) that divides the line segment joining \((x_1,y_1)\) and \((x_2,y_2)\) in the ratio \(m:n\) is given by \(x=\frac{mx_2 + nx_1}{m + n}\) and \(y=\frac{my_2 + ny_1}{m + n}\). Here, the ratio is \(1:2\) (from the beginning to the end), so \(m = 1\), \(n=2\), \((x_1,y_1)=(-1,2)\) (beginning), and \((x,y)=(2,1)\) (the detail point). We need to find \((x_2,y_2)\) (end).

Step2: Solve for \(x_2\)

Using the \(x\)-coordinate formula: \(2=\frac{1\times x_2+2\times(-1)}{1 + 2}\)
Simplify the denominator: \(2=\frac{x_2-2}{3}\)
Multiply both sides by 3: \(6=x_2 - 2\)
Add 2 to both sides: \(x_2=8\)

Step3: Solve for \(y_2\)

Using the \(y\)-coordinate formula: \(1=\frac{1\times y_2+2\times2}{1 + 2}\)
Simplify the denominator: \(1=\frac{y_2 + 4}{3}\)
Multiply both sides by 3: \(3=y_2+4\)
Subtract 4 from both sides: \(y_2=-1\)

Answer:

\((8, -1)\)