Question

which of the following is the correct way to obtain the distance formula? a. $sqrt{(x - 1)^2 + (y - 0)^2}$ b. $sqrt{(x - 0)^2 + (y - 1)^2}$ c. $sqrt{(x - 0)^2 + (y - 0)^2}$ d. $sqrt{(1 - 0)^2 + (1 - 0)^2}$

Explanation:

To solve this, we need to recall the distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\), which is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). However, since the original problem's points are not clearly visible (the image has some unclear text and options), we assume that we need to identify the correct application of the distance formula. Let's assume the two points are, for example, \((1, - 1)\) and \((3,1)\) (just as a sample to show the formula application). The distance between \((x_1,y_1)=(1, - 1)\) and \((x_2,y_2)=(3,1)\) would be \(\sqrt{(3 - 1)^2+(1-(-1))^2}=\sqrt{(3 - 1)^2+(1 + 1)^2}\). But since the original problem's options are partially visible, we need to check the differences in \(x\) and \(y\) coordinates.

Assuming the two points are \((1,-1)\) and \((3,1)\) (matching one of the options' structure), the correct formula would be \(\sqrt{(3 - 1)^2+(1 + 1)^2}\) (if the options have such a form). But due to the unclear text in the image, we can only provide the general approach.

Since the problem is about distance formula, which is in the subfield of Mathematics (Algebra/Geometry).

Step - by - Step (assuming points \((x_1,y_1)\) and \((x_2,y_2)\)):

Step 1: Recall the distance formula

The distance \(d\) between two points \((x_1,y_1)\) and \((x_2,y_2)\) is given by \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).

Step 2: Identify the coordinates from the problem

Suppose the two points are \((x_1,y_1)\) and \((x_2,y_2)\) (from the problem's context). Substitute the values of \(x_1,x_2,y_1,y_2\) into the formula. For example, if \(x_1 = 1,y_1=-1,x_2 = 3,y_2 = 1\), then \(x_2 - x_1=3 - 1\) and \(y_2 - y_1=1-(-1)=1 + 1\), so the distance is \(\sqrt{(3 - 1)^2+(1 + 1)^2}\).

Since the original problem's options are not fully clear, but based on the distance formula, the correct option should be the one that correctly applies \((x_2 - x_1)\) and \((y_2 - y_1)\) in the square - root and square terms. If we assume the points are \((1,-1)\) and \((3,1)\), and the options are:

A. \(\sqrt{(1 - 3)^2+(1 - (-1))^2}\) (equivalent to \(\sqrt{(3 - 1)^2+(1 + 1)^2}\) since \((a - b)^2=(b - a)^2\))

B. \(\sqrt{(3 - 1)^2+(1 + 1)^2}\)

C. \(\sqrt{(1 - 3)^2+(1 + 1)^2}\) (same as A)

D. \(\sqrt{(1 - 3)^2+(1 - (-1))^2}\) (same as A)

The correct formula (using the distance formula) would be \(\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), so if \(x_1 = 1,y_1=-1,x_2 = 3,y_2 = 1\), the correct expression is \(\sqrt{(3 - 1)^2+(1 + 1)^2}\) (assuming option B has this form).

\(\boxed{\sqrt{(3 - 1)^2+(1 + 1)^2}}\) (or the corresponding option from the original problem's choices)

Answer:

To solve this, we need to recall the distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\), which is \(d = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). However, since the original problem's points are not clearly visible (the image has some unclear text and options), we assume that we need to identify the correct application of the distance formula. Let's assume the two points are, for example, \((1, - 1)\) and \((3,1)\) (just as a sample to show the formula application). The distance between \((x_1,y_1)=(1, - 1)\) and \((x_2,y_2)=(3,1)\) would be \(\sqrt{(3 - 1)^2+(1-(-1))^2}=\sqrt{(3 - 1)^2+(1 + 1)^2}\). But since the original problem's options are partially visible, we need to check the differences in \(x\) and \(y\) coordinates.

Assuming the two points are \((1,-1)\) and \((3,1)\) (matching one of the options' structure), the correct formula would be \(\sqrt{(3 - 1)^2+(1 + 1)^2}\) (if the options have such a form). But due to the unclear text in the image, we can only provide the general approach.

Since the problem is about distance formula, which is in the subfield of Mathematics (Algebra/Geometry).

Step - by - Step (assuming points \((x_1,y_1)\) and \((x_2,y_2)\)):

Step 1: Recall the distance formula

The distance \(d\) between two points \((x_1,y_1)\) and \((x_2,y_2)\) is given by \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).

Step 2: Identify the coordinates from the problem

Suppose the two points are \((x_1,y_1)\) and \((x_2,y_2)\) (from the problem's context). Substitute the values of \(x_1,x_2,y_1,y_2\) into the formula. For example, if \(x_1 = 1,y_1=-1,x_2 = 3,y_2 = 1\), then \(x_2 - x_1=3 - 1\) and \(y_2 - y_1=1-(-1)=1 + 1\), so the distance is \(\sqrt{(3 - 1)^2+(1 + 1)^2}\).

Since the original problem's options are not fully clear, but based on the distance formula, the correct option should be the one that correctly applies \((x_2 - x_1)\) and \((y_2 - y_1)\) in the square - root and square terms. If we assume the points are \((1,-1)\) and \((3,1)\), and the options are:

A. \(\sqrt{(1 - 3)^2+(1 - (-1))^2}\) (equivalent to \(\sqrt{(3 - 1)^2+(1 + 1)^2}\) since \((a - b)^2=(b - a)^2\))

B. \(\sqrt{(3 - 1)^2+(1 + 1)^2}\)

C. \(\sqrt{(1 - 3)^2+(1 + 1)^2}\) (same as A)

D. \(\sqrt{(1 - 3)^2+(1 - (-1))^2}\) (same as A)

The correct formula (using the distance formula) would be \(\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), so if \(x_1 = 1,y_1=-1,x_2 = 3,y_2 = 1\), the correct expression is \(\sqrt{(3 - 1)^2+(1 + 1)^2}\) (assuming option B has this form).

\(\boxed{\sqrt{(3 - 1)^2+(1 + 1)^2}}\) (or the corresponding option from the original problem's choices)