Question

Three points are graft on the coordinate grid plus the following points and answer the question .8 is three units to the left of the origin and two units below the X axis point B is five units to the right of the origin and four units but above the X axis points. He is one unit to the right of the origin which point best represents a location that is 4.5 units away from point a.

Answer 1

Let's first determine the coordinates of the points as described:

1. Point A:

   • "three units to the left of the origin" means -3 on the x-axis.
   • "two units below the X-axis" means -2 on the y-axis.
   • Therefore, Point A is at coordinates A(-3, -2).

2. Point B:

   • "five units to the right of the origin" gives us 5 on the x-axis.
   • "four units above the X-axis" means +4 on the y-axis.
   • Thus, Point B is at coordinates B(5, 4).

3. Point C:

   • "one unit to the right of the origin" gives us 1 on the x-axis.
   • No vertical position is mentioned, so we can assume it is on the x-axis (y=0).
   • Therefore, Point C is at coordinates C(1, 0).

Now, we need to find a point that is 4.5 units away from Point A(-3, -2). The distance formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is given by:

$$d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$$

Substituting Point A's coordinates into the formula, we need the distance $d$ to be 4.5 units. Thus, we have:

$$4.5=\sqrt{(x+3{)}^2+(y+2{)}^2}$$

Squaring both sides gives us:

$$20.25=(x+3{)}^2+(y+2{)}^2$$

Now, we can find multiple points that satisfy this equation.

To find potential points, we can start with some easy values for $x$ and then calculate $y$.

1. If $x=-3$: $$(y+2{)}^2=20.25⟹y+2=\pm 4.5⟹y=2.5\text{ or }-6.5$$ Points: (-3, 2.5) and (-3, -6.5)

2. If $x=1$: $$(1+3{)}^2+(y+2{)}^2=20.25⟹16+(y+2{)}^2=20.25⟹(y+2{)}^2=4.25$$ $⟹y+2=\pm \sqrt{4.25}⟹y=-2+\sqrt{4.25}$ or $-2-\sqrt{4.25}$

   Approximating $\sqrt{4.25}\approx 2.06$:

   • $y\approx -2+2.06\approx 0.06$
   • $y\approx -2-2.06\approx -4.06$

   Point: (1, 0.06) and (1, -4.06)

3. If $x=0$: $$(0+3{)}^2+(y+2{)}^2=20.25\to 9+(y+2{)}^2=20.25⟹(y+2{)}^2=11.25$$ $⟹y+2=\pm \sqrt{11.25}⟹y=-2+\sqrt{11.25}$ or $-2-\sqrt{11.25}$

   Approximating $\sqrt{11.25}\approx 3.35$:

   • $y\approx -2+3.35\approx 1.35$
   • $y\approx -2-3.35\approx -5.35$

   Point: (0, 1.35) and (0, -5.35)

These calculations give us multiple points that are 4.5 units away from point A (-3, -2).

Some of these points could be:

• (-3, 2.5)
• (-3, -6.5)
• (1, 0.06)
• (1, -4.06)
• (0, 1.35)
• (0, -5.35)

You can choose any of these points based on your specific requirement or context you have in your question.