Annie is creating a stencil for her artwork using a coordinate plane. The beginning of the left edge of the stencil falls at (1, -1). She wants to align an important detail on the left edge of her stencil at (3, 0). This point is 1/3 of the way to the end of the stencil.

Where is the end of the stencil located?

A. (1, 5.075)
B. (2.5, -0.25)
C. (6, 2)
D. (9, 3)



Answer :

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].