Given the Distance Formula: D= (x2−x1)2+(y2 − y1)2 , what is the distance between the points (1, 2) and (4, 6)?(1 point)

Question

Responses

D = 10
D = 10

Distance cannot be found.
Distance cannot be found.

D = 5
D = 5

D = 7
D = 7

Answer 1

To calculate the distance between the points \((1, 2)\) and \((4, 6)\) using the distance formula \(D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\), we can follow these steps:

1. Assign the coordinates:

   • \((x_1, y_1) = (1, 2)\)
   • \((x_2, y_2) = (4, 6)\)

2. Substitute the coordinates into the formula: \[ D = \sqrt{(4 - 1)^2 + (6 - 2)^2} \]

3. Calculate the differences: \[ D = \sqrt{(3)^2 + (4)^2} \]

4. Square the differences: \[ D = \sqrt{9 + 16} \]

5. Sum the results: \[ D = \sqrt{25} \]

6. Finally, take the square root: \[ D = 5 \]

Therefore, the distance between the points \((1, 2)\) and \((4, 6)\) is \(D = 5\).

So the correct response is: D = 5.

Answer 2

What is the formula to find the midpoint between two points (x2 , y2) and (x1 , y1) ?(1 point)
Responses

(x1+x22 , y1−y22)(x1+x22 , y1−y22)

(x1+x22 , y1+y22)(x1+x22 , y1+y22)

(x1−x22 , y1−y22)(x1−x22 , y1−y22)

(x1−x22 , y1+y22)

Answer 3

The formula to find the midpoint \(M\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

This formula calculates the average of the x-coordinates and the average of the y-coordinates of the two points.

None of the responses provided are correctly formatted to indicate the midpoint formula correctly. The correct formulation would be:

\[ \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

However, based on your provided options, the closest equivalent expression could resemble \((x_1+x_2)/2, (y_1+y_2)/2\).

If you have any further options or clarifications, let me know!

Answer 4

Given points (1,−2) and (7,6), how do you calculate the x-coordinate of the midpoint?(1 point)
Responses

−2−62the fraction with numerator negative 2 minus 6 and denominator 2

6+−22the fraction with numerator 6 plus negative 2 and denominator 2

1+72the fraction with numerator 1 plus 7 and denominator 2

1−72the fraction with numerator 1 minus 7 and denominator 2

Answer 5

To calculate the x-coordinate of the midpoint between the points \((1, -2)\) and \((7, 6)\), you can use the formula for the midpoint. Specifically, the x-coordinate of the midpoint \(M\) is given by:

\[ M_x = \frac{x_1 + x_2}{2} \]

In this case, \(x_1 = 1\) and \(x_2 = 7\). Substituting these values into the formula gives:

\[ M_x = \frac{1 + 7}{2} \] \[ M_x = \frac{8}{2} = 4 \]

Looking at the responses you provided:

1. \(−2−62\) (the fraction with numerator negative 2 minus 6 and denominator 2)
2. \(6+−22\) (the fraction with numerator 6 plus negative 2 and denominator 2)
3. \(1+72\) (the fraction with numerator 1 plus 7 and denominator 2)
4. \(1−72\) (the fraction with numerator 1 minus 7 and denominator 2)

The correct expression for the x-coordinate of the midpoint between the points (1,−2) and (7,6) corresponds to the response:

3. \(1+72\) (the fraction with numerator 1 plus 7 and denominator 2)

This is the appropriate way to express the calculation to find the x-coordinate of the midpoint.

Answer 6

Given points (1,−2) and (7,6), how do you calculate the y-coordinate of the midpoint?(1 point)
Responses

1 + 721 + 72

1 − 721 − 72

−2 − 62−2 − 62

−2 + 62

Answer 7

To calculate the y-coordinate of the midpoint between the points \((1, -2)\) and \((7, 6)\), you use the formula for the midpoint, specifically for the y-coordinate:

\[ M_y = \frac{y_1 + y_2}{2} \]

In this case, \(y_1 = -2\) and \(y_2 = 6\). Substituting these values into the formula gives:

\[ M_y = \frac{-2 + 6}{2} \] \[ M_y = \frac{4}{2} = 2 \]

Now, looking at the responses you provided:

1. \(1 + 72\)
2. \(1 − 72\)
3. \(−2 − 62\)
4. \(−2 + 62\)

The correct expression for the y-coordinate of the midpoint is:

4. \(−2 + 62\)

This correctly represents the calculation to find the y-coordinate of the midpoint as \(\frac{-2 + 6}{2}\).

Answer 8

Given points (0,0) and (10,10), what is the midpoint?(1 point)
Responses

(5 , −5)(5 , −5)

(−5 , −5)(−5 , −5)

(5 , 5)(5 , 5)

(0 , 5)(0 , 5)

Answer 9

To find the midpoint between the points \((0, 0)\) and \((10, 10)\), you can use the midpoint formula:

\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

Substituting the coordinates of the given points \((x_1, y_1) = (0, 0)\) and \((x_2, y_2) = (10, 10)\):

1. For the x-coordinate: \[ M_x = \frac{0 + 10}{2} = \frac{10}{2} = 5 \]

2. For the y-coordinate: \[ M_y = \frac{0 + 10}{2} = \frac{10}{2} = 5 \]

Thus, the midpoint \(M\) is:

\[ M = (5, 5) \]

Now, looking at the responses you provided:

1. \((5, −5)\)
2. \((-5, −5)\)
3. \((5, 5)\)
4. \((0, 5)\)

The correct response is:

\((5, 5)\).