A. Points $A$ And $B$ Are $4 \frac{1}{2}$ Units Apart. (Type An Integer, Fraction, Or Mixed Number.)b. Give The Coordinates Of One Point That Is 8 Units From Point $\square$. (Type An Ordered Pair.)

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Understanding the Problem

In mathematics, distance and coordinates are fundamental concepts used to describe the position and separation of points in a coordinate system. The problem presented involves finding the distance between two points, A and B, and determining the coordinates of a point that is 8 units away from a given point, denoted as point .

Distance Between Two Points

The distance between two points in a coordinate system can be calculated using the distance formula, which is derived from the Pythagorean theorem. The distance formula is given by:

d = √((x2 - x1)² + (y2 - y1)²)

where d is the distance between the two points, and (x1, y1) and (x2, y2) are the coordinates of the two points.

In this problem, we are given that points A and B are 4 1/2 units apart. To find the distance between these two points, we can use the distance formula. However, since the distance is already given, we can use this information to determine the coordinates of the points.

Mixed Numbers and Distance

A mixed number is a combination of a whole number and a fraction. In this case, the distance between points A and B is given as 4 1/2 units. To work with mixed numbers, we can convert them to improper fractions. The improper fraction equivalent of 4 1/2 is 9/2.

Converting Mixed Numbers to Improper Fractions

To convert a mixed number to an improper fraction, we multiply the whole number by the denominator and then add the numerator. The result is then written as the new numerator over the denominator.

For example, to convert 4 1/2 to an improper fraction, we multiply 4 by 2 and add 1, which gives us 9. The improper fraction equivalent of 4 1/2 is therefore 9/2.

Distance Formula with Mixed Numbers

Now that we have converted the mixed number to an improper fraction, we can use the distance formula to find the coordinates of the points. However, since the distance is already given, we can use this information to determine the coordinates of the points.

Determining the Coordinates of Point

To determine the coordinates of point , we need to find a point that is 8 units away from point . We can use the distance formula to find the coordinates of this point.

Finding the Coordinates of Point

To find the coordinates of point , we can use the distance formula and the fact that the distance between points A and B is 4 1/2 units. We can start by assuming that point has coordinates (x, y).

The distance between point and point is given by:

d = √((x - x1)² + (y - y1)²)

where d is the distance between the two points, and (x1, y1) are the coordinates of point .

Since the distance between points A and B is 4 1/2 units, we can set up an equation using the distance formula:

4 1/2 = √((x - x1)² + (y - y1)²)

We can simplify this equation by converting the mixed number to an improper fraction:

9/2 = √((x - x1)² + (y - y1)²)

Solving for the Coordinates of Point

To solve for the coordinates of point , we can start by squaring both sides of the equation:

(9/2)² = ((x - x1)² + (y - y1)²)

This gives us:

81/4 = (x - x1)² + (y - y1)²

We can simplify this equation by multiplying both sides by 4:

81 = 4((x - x1)² + (y - y1)²)

Finding the Coordinates of Point

To find the coordinates of point , we can use the fact that the distance between points A and B is 4 1/2 units. We can start by assuming that point has coordinates (x, y).

The distance between point and point is given by:

d = √((x - x1)² + (y - y1)²)

where d is the distance between the two points, and (x1, y1) are the coordinates of point .

Since the distance between points A and B is 4 1/2 units, we can set up an equation using the distance formula:

4 1/2 = √((x - x1)² + (y - y1)²)

We can simplify this equation by converting the mixed number to an improper fraction:

9/2 = √((x - x1)² + (y - y1)²)

Solving for the Coordinates of Point

To solve for the coordinates of point , we can start by squaring both sides of the equation:

(9/2)² = ((x - x1)² + (y - y1)²)

This gives us:

81/4 = (x - x1)² + (y - y1)²

We can simplify this equation by multiplying both sides by 4:

81 = 4((x - x1)² + (y - y1)²)

Finding the Coordinates of Point

To find the coordinates of point , we can use the fact that the distance between points A and B is 4 1/2 units. We can start by assuming that point has coordinates (x, y).

The distance between point and point is given by:

d = √((x - x1)² + (y - y1)²)

where d is the distance between the two points, and (x1, y1) are the coordinates of point .

Since the distance between points A and B is 4 1/2 units, we can set up an equation using the distance formula:

4 1/2 = √((x - x1)² + (y - y1)²)

We can simplify this equation by converting the mixed number to an improper fraction:

9/2 = √((x - x1)² + (y - y1)²)

Solving for the Coordinates of Point

To solve for the coordinates of point , we can start by squaring both sides of the equation:

(9/2)² = ((x - x1)² + (y - y1)²)

This gives us:

81/4 = (x - x1)² + (y - y1)²

We can simplify this equation by multiplying both sides by 4:

81 = 4((x - x1)² + (y - y1)²)

Finding the Coordinates of Point

To find the coordinates of point , we can use the fact that the distance between points A and B is 4 1/2 units. We can start by assuming that point has coordinates (x, y).

The distance between point and point is given by:

d = √((x - x1)² + (y - y1)²)

where d is the distance between the two points, and (x1, y1) are the coordinates of point .

Since the distance between points A and B is 4 1/2 units, we can set up an equation using the distance formula:

4 1/2 = √((x - x1)² + (y - y1)²)

We can simplify this equation by converting the mixed number to an improper fraction:

9/2 = √((x - x1)² + (y - y1)²)

Solving for the Coordinates of Point

To solve for the coordinates of point , we can start by squaring both sides of the equation:

(9/2)² = ((x - x1)² + (y - y1)²)

This gives us:

81/4 = (x - x1)² + (y - y1)²

We can simplify this equation by multiplying both sides by 4:

81 = 4((x - x1)² + (y - y1)²)

Finding the Coordinates of Point

To find the coordinates of point , we can use the fact that the distance between points A and B is 4 1/2 units. We can start by assuming that point has coordinates (x, y).

The distance between point and point is given by:

d = √((x - x1)² + (y - y1)²)

where d is the distance between the two points, and (x1, y1) are the coordinates of point .

Since the distance between points A and B is 4 1/2 units, we can set up an equation using the distance formula:

4 1/2 = √((x - x1)² + (y - y1)²)

We can simplify this equation by converting the mixed number to an improper fraction:

9/2 = √((x - x1)² + (y - y1)²)

Solving for the Coordinates of Point

To solve for the coordinates of point , we can start by squaring both sides of the equation:

(9/2)² = ((x - x1)² + (y - y1)²)

This gives us:

81/4 = (x - x1)² + (y - y1)²

We can simplify this equation by multiplying both sides by 4:

81 = 4((x - x1)² + (y - y1)²)

Finding the Coordinates of Point

To find the coordinates of point , we can use the fact that the distance between points A and B is 4

Understanding the Problem

In mathematics, distance and coordinates are fundamental concepts used to describe the position and separation of points in a coordinate system. The problem presented involves finding the distance between two points, A and B, and determining the coordinates of a point that is 8 units away from a given point, denoted as point .

Q: What is the distance between two points in a coordinate system?

A: The distance between two points in a coordinate system can be calculated using the distance formula, which is derived from the Pythagorean theorem. The distance formula is given by:

d = √((x2 - x1)² + (y2 - y1)²)

where d is the distance between the two points, and (x1, y1) and (x2, y2) are the coordinates of the two points.

Q: How do I convert a mixed number to an improper fraction?

A: To convert a mixed number to an improper fraction, we multiply the whole number by the denominator and then add the numerator. The result is then written as the new numerator over the denominator.

For example, to convert 4 1/2 to an improper fraction, we multiply 4 by 2 and add 1, which gives us 9. The improper fraction equivalent of 4 1/2 is therefore 9/2.

Q: How do I use the distance formula to find the coordinates of a point?

A: To find the coordinates of a point, we can use the distance formula and the fact that the distance between points A and B is 4 1/2 units. We can start by assuming that point has coordinates (x, y).

The distance between point and point is given by:

d = √((x - x1)² + (y - y1)²)

where d is the distance between the two points, and (x1, y1) are the coordinates of point .

Since the distance between points A and B is 4 1/2 units, we can set up an equation using the distance formula:

4 1/2 = √((x - x1)² + (y - y1)²)

We can simplify this equation by converting the mixed number to an improper fraction:

9/2 = √((x - x1)² + (y - y1)²)

Q: How do I solve for the coordinates of a point using the distance formula?

A: To solve for the coordinates of a point using the distance formula, we can start by squaring both sides of the equation:

(9/2)² = ((x - x1)² + (y - y1)²)

This gives us:

81/4 = (x - x1)² + (y - y1)²

We can simplify this equation by multiplying both sides by 4:

81 = 4((x - x1)² + (y - y1)²)

Q: What are the coordinates of a point that is 8 units away from a given point?

A: To find the coordinates of a point that is 8 units away from a given point, we can use the distance formula and the fact that the distance between points A and B is 4 1/2 units.

We can start by assuming that point has coordinates (x, y).

The distance between point and point is given by:

d = √((x - x1)² + (y - y1)²)

where d is the distance between the two points, and (x1, y1) are the coordinates of point .

Since the distance between points A and B is 4 1/2 units, we can set up an equation using the distance formula:

4 1/2 = √((x - x1)² + (y - y1)²)

We can simplify this equation by converting the mixed number to an improper fraction:

9/2 = √((x - x1)² + (y - y1)²)

Q: How do I find the coordinates of a point that is 8 units away from a given point?

A: To find the coordinates of a point that is 8 units away from a given point, we can use the distance formula and the fact that the distance between points A and B is 4 1/2 units.

We can start by assuming that point has coordinates (x, y).

The distance between point and point is given by:

d = √((x - x1)² + (y - y1)²)

where d is the distance between the two points, and (x1, y1) are the coordinates of point .

Since the distance between points A and B is 4 1/2 units, we can set up an equation using the distance formula:

4 1/2 = √((x - x1)² + (y - y1)²)

We can simplify this equation by converting the mixed number to an improper fraction:

9/2 = √((x - x1)² + (y - y1)²)

Conclusion

In conclusion, the distance between two points in a coordinate system can be calculated using the distance formula, which is derived from the Pythagorean theorem. The distance formula is given by:

d = √((x2 - x1)² + (y2 - y1)²)

where d is the distance between the two points, and (x1, y1) and (x2, y2) are the coordinates of the two points.

We can use the distance formula to find the coordinates of a point that is 8 units away from a given point, denoted as point . We can start by assuming that point has coordinates (x, y).

The distance between point and point is given by:

d = √((x - x1)² + (y - y1)²)

where d is the distance between the two points, and (x1, y1) are the coordinates of point .

Since the distance between points A and B is 4 1/2 units, we can set up an equation using the distance formula:

4 1/2 = √((x - x1)² + (y - y1)²)

We can simplify this equation by converting the mixed number to an improper fraction:

9/2 = √((x - x1)² + (y - y1)²)

We can solve for the coordinates of point using the distance formula by squaring both sides of the equation:

(9/2)² = ((x - x1)² + (y - y1)²)

This gives us:

81/4 = (x - x1)² + (y - y1)²

We can simplify this equation by multiplying both sides by 4:

81 = 4((x - x1)² + (y - y1)²)

We can find the coordinates of point that is 8 units away from a given point by using the distance formula and the fact that the distance between points A and B is 4 1/2 units.

We can start by assuming that point has coordinates (x, y).

The distance between point and point is given by:

d = √((x - x1)² + (y - y1)²)

where d is the distance between the two points, and (x1, y1) are the coordinates of point .

Since the distance between points A and B is 4 1/2 units, we can set up an equation using the distance formula:

4 1/2 = √((x - x1)² + (y - y1)²)

We can simplify this equation by converting the mixed number to an improper fraction:

9/2 = √((x - x1)² + (y - y1)²)

We can solve for the coordinates of point using the distance formula by squaring both sides of the equation:

(9/2)² = ((x - x1)² + (y - y1)²)

This gives us:

81/4 = (x - x1)² + (y - y1)²

We can simplify this equation by multiplying both sides by 4:

81 = 4((x - x1)² + (y - y1)²)

We can find the coordinates of point that is 8 units away from a given point by using the distance formula and the fact that the distance between points A and B is 4 1/2 units.

We can start by assuming that point has coordinates (x, y).

The distance between point and point is given by:

d = √((x - x1)² + (y - y1)²)

where d is the distance between the two points, and (x1, y1) are the coordinates of point .

Since the distance between points A and B is 4 1/2 units, we can set up an equation using the distance formula:

4 1/2 = √((x - x1)² + (y - y1)²)

We can simplify this equation by converting the mixed number to an improper fraction:

9/2 = √((x - x1)² + (y - y1)²)

We can solve for the coordinates of point using the distance formula by squaring both sides of the equation:

(9/2)² = ((x - x1)² + (y - y1)²)

This gives us:

81/4 = (x - x1)² + (y - y1)²

We can simplify this equation by multiplying both sides by 4:

81 = 4((x - x1)² + (