find the distance d (P_1 & P_2) between points P_1 and P_1.

P_1=(-3,3)
P_2=(1,5)

Find the midpoint of the line segment joining the points P_1 and P_2.
P_1=(1,1)
P_2=(-3,3)

Please show all work

To find the distance between the two points, we can use the distance formula:

d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Given the coordinates of the two points, we can plug them into the formula:

P_1 = (-3, 3)
P_2 = (1, 5)

d = sqrt((1 - (-3))^2 + (5 - 3)^2)

d = sqrt((1 + 3)^2 + (5 - 3)^2)

d = sqrt(4^2 + 2^2)

d = sqrt(16 + 4)

d = sqrt(20)

So, the distance between the two points is √20.

Now, to find the midpoint of the line segment joining the two points, we can use the midpoint formula:

M = ((x1 + x2)/2, (y1 + y2)/2)

Given the coordinates of the two points, we can plug them into the formula:

P_1 = (1, 1)
P_2 = (-3, 3)

M = ((1 + (-3))/2, (1 + 3)/2)

M = ((1 - 3)/2, (1 + 3)/2)

M = (-2/2, 4/2)

M = (-1, 2)

So, the midpoint of the line segment joining the two points is (-1, 2).

To find the distance d between points P_1 and P_2, we can use the distance formula:

d = √((x₂ - x₁)² + (y₂ - y₁)²)

Given:
P_1 = (-3, 3)
P_2 = (1, 5)

Let's calculate the distance:

d = √((1 - (-3))² + (5 - 3)²)
= √((1 + 3)² + (5 - 3)²)
= √(4² + 2²)
= √(16 + 4)
= √20
= 2√5

Therefore, the distance between points P_1 and P_2 is 2√5.

To find the midpoint of the line segment joining points P_1 and P_2, we can use the midpoint formula:

Midpoint (M) = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)

Given:
P_1 = (1, 1)
P_2 = (-3, 3)

Let's calculate the midpoint:

x-coordinate of M = (1 + (-3)) / 2 = -2 / 2 = -1
y-coordinate of M = (1 + 3) / 2 = 4 / 2 = 2

Therefore, the midpoint of the line segment joining points P_1 and P_2 is M(-1, 2).

To find the distance \( d \) between two points \( P_1 \) and \( P_2 \), we can use the distance formula:

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

where \( (x_1, y_1) \) are the coordinates of \( P_1 \) and \( (x_2, y_2) \) are the coordinates of \( P_2 \).

Given:
\( P_1 = (-3, 3) \)
\( P_2 = (1, 5) \)

Substituting the values into the distance formula:
\[ d = \sqrt{{(1 - (-3))^2 + (5 - 3)^2}} \]
\[ d = \sqrt{{(1 + 3)^2 + (5 - 3)^2}} \]
\[ d = \sqrt{{4^2 + 2^2}} \]
\[ d = \sqrt{{16 + 4}} \]
\[ d = \sqrt{{20}} \]
\[ d = 2 \sqrt{{5}} \]

Therefore, the distance \( d \) between \( P_1 \) and \( P_2 \) is \( 2 \sqrt{{5}} \).

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To find the midpoint \( M \) of the line segment joining the points \( P_1 \) and \( P_2 \), we can use the midpoint formula:

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

where \( (x_1, y_1) \) are the coordinates of \( P_1 \) and \( (x_2, y_2) \) are the coordinates of \( P_2 \).

Given:
\( P_1 = (1, 1) \)
\( P_2 = (-3, 3) \)

Substituting the values into the midpoint formula:
\[ M = \left(\frac{{1 + (-3)}}{2}, \frac{{1 + 3}}{2}\right) \]
\[ M = \left(\frac{{-2}}{2}, \frac{{4}}{2}\right) \]
\[ M = (-1, 2) \]

Therefore, the midpoint \( M \) of the line segment joining \( P_1 \) and \( P_2 \) is (-1, 2).