Sure! Let's solve the given question step-by-step.
### Given:
Vertices of the parallelogram ABCD:
- A: (1, 2)
- B: (3, -1)
- C: (-1, -1)
We need to find:
(i) The coordinates of D
(ii) The length of AC
### Solution:
#### (i) Coordinates of D:
We know that in a parallelogram, the diagonals bisect each other. This means that the midpoints of both diagonals are the same. To find the coordinates of D, we'll use this property.
1. Calculate the midpoint of diagonal AC:
The midpoint formula for coordinates [tex]\( (x_1, y_1) \)[/tex] and [tex]\( (x_2, y_2) \)[/tex] is given by:
[tex]\[
\text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\][/tex]
Substituting the coordinates of A and C:
[tex]\[
\text{Midpoint of AC} = \left( \frac{1 + (-1)}{2}, \frac{2 + (-1)}{2} \right) = \left( \frac{0}{2}, \frac{1}{2} \right) = (0, 0.5)
\][/tex]
2. Find coordinates of D using the midpoint of BD:
Let's denote the coordinates of D as [tex]\( (x, y) \)[/tex]. The midpoint of diagonal BD must be the same as the midpoint of AC.
[tex]\[
\left( \frac{3 + x}{2}, \frac{-1 + y}{2} \right) = (0, 0.5)
\][/tex]
Setting up the equations for the x and y coordinates:
[tex]\[
\frac{3 + x}{2} = 0 \quad \Rightarrow \quad 3 + x = 0 \quad \Rightarrow \quad x = -3
\][/tex]
[tex]\[
\frac{-1 + y}{2} = 0.5 \quad \Rightarrow \quad -1 + y = 1 \quad \Rightarrow \quad y = 2
\][/tex]
Therefore, the coordinates of D are [tex]\( (-3, 2) \)[/tex].
#### (ii) Length of AC:
To find the length of AC, we use the distance formula. For points [tex]\( (x_1, y_1) \)[/tex] and [tex]\( (x_2, y_2) \)[/tex], the distance [tex]\( d \)[/tex] is given by:
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Substituting the coordinates of A and C:
[tex]\[
\text{Distance AC} = \sqrt{((-1) - 1)^2 + ((-1) - 2)^2} = \sqrt{(-2)^2 + (-3)^2} = \sqrt{4 + 9} = \sqrt{13}
\][/tex]
Thus, the length of AC is approximately 3.6056.
### Summary:
(i) The coordinates of D are [tex]\( (-3, 2) \)[/tex].
(ii) The length of AC is [tex]\(\sqrt{13} \approx 3.6056\)[/tex].
