Answer :
Absolutely, let's tackle each part of the question step by step.
### Part (a): Find the exact distance between the points
The coordinates of the points are:
- [tex]\( A(-1, 4) \)[/tex]
- [tex]\( B(4, -1) \)[/tex]
To find the distance between the points [tex]\( A \)[/tex] and [tex]\( B \)[/tex], we use the distance formula, which is:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Substituting the given coordinates into the formula:
[tex]\[ d = \sqrt{(4 - (-1))^2 + (-1 - 4)^2} \][/tex]
Simplify the expressions inside the square root:
[tex]\[ d = \sqrt{(4 + 1)^2 + (-1 - 4)^2} \][/tex]
[tex]\[ d = \sqrt{5^2 + (-5)^2} \][/tex]
[tex]\[ d = \sqrt{25 + 25} \][/tex]
[tex]\[ d = \sqrt{50} \][/tex]
[tex]\[ d = \sqrt{25 \times 2} \][/tex]
[tex]\[ d = 5\sqrt{2} \][/tex]
Converting it to a decimal approximation:
[tex]\[ d \approx 7.0710678118654755 \][/tex]
So, the exact distance between the points is:
[tex]\[ 5\sqrt{2} \quad \text{or approximately} \quad 7.0710678118654755 \][/tex]
### Part (b): Find the midpoint of the line segment whose endpoints are the given points
The midpoint [tex]\( M \)[/tex] of a line segment with endpoints [tex]\( (x_1, y_1) \)[/tex] and [tex]\( (x_2, y_2) \)[/tex] is given by the midpoint formula:
[tex]\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \][/tex]
Substituting the given coordinates into the formula:
[tex]\[ M = \left( \frac{-1 + 4}{2}, \frac{4 + (-1)}{2} \right) \][/tex]
Simplify the expressions:
[tex]\[ M = \left( \frac{3}{2}, \frac{3}{2} \right) \][/tex]
Thus, the coordinates of the midpoint are:
[tex]\[ M = \left( 1.5, 1.5 \right) \][/tex]
### Summary of Results
(a) The exact distance between the points [tex]\( (-1, 4) \)[/tex] and [tex]\( (4, -1) \)[/tex] is [tex]\( 5\sqrt{2} \approx 7.0710678118654755 \)[/tex].
(b) The midpoint of the line segment with endpoints [tex]\( (-1, 4) \)[/tex] and [tex]\( (4, -1) \)[/tex] is [tex]\( (1.5, 1.5) \)[/tex].
### Part (a): Find the exact distance between the points
The coordinates of the points are:
- [tex]\( A(-1, 4) \)[/tex]
- [tex]\( B(4, -1) \)[/tex]
To find the distance between the points [tex]\( A \)[/tex] and [tex]\( B \)[/tex], we use the distance formula, which is:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Substituting the given coordinates into the formula:
[tex]\[ d = \sqrt{(4 - (-1))^2 + (-1 - 4)^2} \][/tex]
Simplify the expressions inside the square root:
[tex]\[ d = \sqrt{(4 + 1)^2 + (-1 - 4)^2} \][/tex]
[tex]\[ d = \sqrt{5^2 + (-5)^2} \][/tex]
[tex]\[ d = \sqrt{25 + 25} \][/tex]
[tex]\[ d = \sqrt{50} \][/tex]
[tex]\[ d = \sqrt{25 \times 2} \][/tex]
[tex]\[ d = 5\sqrt{2} \][/tex]
Converting it to a decimal approximation:
[tex]\[ d \approx 7.0710678118654755 \][/tex]
So, the exact distance between the points is:
[tex]\[ 5\sqrt{2} \quad \text{or approximately} \quad 7.0710678118654755 \][/tex]
### Part (b): Find the midpoint of the line segment whose endpoints are the given points
The midpoint [tex]\( M \)[/tex] of a line segment with endpoints [tex]\( (x_1, y_1) \)[/tex] and [tex]\( (x_2, y_2) \)[/tex] is given by the midpoint formula:
[tex]\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \][/tex]
Substituting the given coordinates into the formula:
[tex]\[ M = \left( \frac{-1 + 4}{2}, \frac{4 + (-1)}{2} \right) \][/tex]
Simplify the expressions:
[tex]\[ M = \left( \frac{3}{2}, \frac{3}{2} \right) \][/tex]
Thus, the coordinates of the midpoint are:
[tex]\[ M = \left( 1.5, 1.5 \right) \][/tex]
### Summary of Results
(a) The exact distance between the points [tex]\( (-1, 4) \)[/tex] and [tex]\( (4, -1) \)[/tex] is [tex]\( 5\sqrt{2} \approx 7.0710678118654755 \)[/tex].
(b) The midpoint of the line segment with endpoints [tex]\( (-1, 4) \)[/tex] and [tex]\( (4, -1) \)[/tex] is [tex]\( (1.5, 1.5) \)[/tex].
