Sure, let's break this down step-by-step:
1. Identify the Starting and Target Points:
- The starting point (let's call it [tex]\( A \)[/tex]) is [tex]\((1, -1)\)[/tex].
- The important detail point (let's call it [tex]\( B \)[/tex]) is [tex]\((3, 0)\)[/tex].
2. Understand the Fraction of the Journey:
- The point [tex]\( B \)[/tex] is [tex]\( \frac{1}{3} \)[/tex] of the way to the end point (let's call it [tex]\( C \)[/tex]).
3. Calculate the Full Distance Needed:
- If [tex]\(\frac{1}{3}\)[/tex] of the way is from [tex]\(A\)[/tex] to [tex]\(B\)[/tex], to find the whole journey (full distance) we need to travel [tex]\(3\)[/tex] times the distance from [tex]\(A\)[/tex] to [tex]\(B\)[/tex].
4. Find the Distance Between [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
- Change in [tex]\( x \)[/tex] direction: [tex]\( 3 - 1 = 2 \)[/tex]
- Change in [tex]\( y \)[/tex] direction: [tex]\( 0 - (-1) = 1 \)[/tex]
5. Calculate the Full Distance (3 times this distance):
- Full distance in [tex]\( x \)[/tex] direction: [tex]\( 2 \times 3 = 6 \)[/tex]
- Full distance in [tex]\( y \)[/tex] direction: [tex]\( 1 \times 3 = 3 \)[/tex]
6. Identify the End Point:
- Start at point [tex]\( A \)[/tex] [tex]\((1, -1)\)[/tex]
- Add the full distances calculated to find point [tex]\( C \)[/tex]:
- [tex]\( x \)[/tex] coordinate: [tex]\( 1 + 6 = 7 \)[/tex]
- [tex]\( y \)[/tex] coordinate: [tex]\( -1 + 3 = 2 \)[/tex]
So, the coordinates for the end of the stencil are [tex]\( (7, 2) \)[/tex].
The correct answer to the question is [tex]\((7, 2)\)[/tex], which aligns with option [tex]\((6, 2)\)[/tex].
Let's note that there might be a mistake in the options given, as option [tex]\((6, 2)\)[/tex] seems closest but slightly off, perhaps due to a typo in the question. Regardless, based on the calculations, the end of the stencil should be at [tex]\( (7, 2) \)[/tex].
