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Sagot :
Certainly! Let's solve the problem step-by-step to determine the coordinates of point D in the parallelogram formed by points A(-2, 4), B(1, 3), and C(4, -1).
### Step 1: Calculate Vector AB
First, we need to find the vector from point A to point B, denoted as [tex]\( \overrightarrow{AB} \)[/tex].
The coordinates of A are (-2, 4) and the coordinates of B are (1, 3).
[tex]\[ \overrightarrow{AB} = B - A = (1 - (-2), 3 - 4) = (1 + 2, 3 - 4) = (3, -1) \][/tex]
### Step 2: Calculate Vector AC
Next, we need to find the vector from point A to point C, denoted as [tex]\( \overrightarrow{AC} \)[/tex].
The coordinates of C are (4, -1).
[tex]\[ \overrightarrow{AC} = C - A = (4 - (-2), -1 - 4) = (4 + 2, -1 - 4) = (6, -5) \][/tex]
### Step 3: Find Point D Using the Parallelogram Property
In a parallelogram, the vector from one vertex to the opposite vertex is the sum of two adjacent vectors. Thus:
[tex]\[ \overrightarrow{AD} = \overrightarrow{AB} + \overrightarrow{AC} \][/tex]
However, another method to find point D directly is to use the property that opposite sides are equal. Point D can be found by adding vector [tex]\( \overrightarrow{AB} \)[/tex] to point C (or equivalently, adding vector [tex]\( \overrightarrow{AC} \)[/tex] to point B).
Using point C and vector [tex]\( \overrightarrow{AB} \)[/tex]:
[tex]\[ D = C + \overrightarrow{AB} = (4, -1) + (3, -1) = (4 + 3, -1 - 1) = (7, -2) \][/tex]
Alternatively, you could use:
[tex]\[ D = B + \overrightarrow{AC} = (1, 3) + (6, -5) = (1 + 6, 3 - 5) = (7, -2) \][/tex]
However, this result (7, -2) does not match any of the given options in the question. Let’s recheck another method by midpoint property:
### Step 4: Alternative Method Using Midpoints
In a parallelogram, the diagonals bisect each other. So if M is the midpoint of both diagonals AC and BD:
[tex]\[ M = \frac{A + C}{2} = \frac{(-2, 4) + (4, -1)}{2} = \frac{(2, 3)}{2} = (1, 1.5) \][/tex]
Given M should also be midpoint of BD:
[tex]\[ M = \frac{B + D}{2} \implies (1, 1.5) = \frac{(1, 3) + D}{2} \][/tex]
Solving for D:
[tex]\[ 2 \times (1, 1.5) = (1, 3) + D \][/tex]
[tex]\[ (2, 3) = (1, 3) + D \][/tex]
So,
[tex]\[ D = (2, 3) - (1, 3) = (1, 0) \][/tex]
The correct coordinates of point D are:
[tex]\[ \boxed{(1, 0)} \][/tex]
Thus, the correct option is:
D. (1,0)
### Step 1: Calculate Vector AB
First, we need to find the vector from point A to point B, denoted as [tex]\( \overrightarrow{AB} \)[/tex].
The coordinates of A are (-2, 4) and the coordinates of B are (1, 3).
[tex]\[ \overrightarrow{AB} = B - A = (1 - (-2), 3 - 4) = (1 + 2, 3 - 4) = (3, -1) \][/tex]
### Step 2: Calculate Vector AC
Next, we need to find the vector from point A to point C, denoted as [tex]\( \overrightarrow{AC} \)[/tex].
The coordinates of C are (4, -1).
[tex]\[ \overrightarrow{AC} = C - A = (4 - (-2), -1 - 4) = (4 + 2, -1 - 4) = (6, -5) \][/tex]
### Step 3: Find Point D Using the Parallelogram Property
In a parallelogram, the vector from one vertex to the opposite vertex is the sum of two adjacent vectors. Thus:
[tex]\[ \overrightarrow{AD} = \overrightarrow{AB} + \overrightarrow{AC} \][/tex]
However, another method to find point D directly is to use the property that opposite sides are equal. Point D can be found by adding vector [tex]\( \overrightarrow{AB} \)[/tex] to point C (or equivalently, adding vector [tex]\( \overrightarrow{AC} \)[/tex] to point B).
Using point C and vector [tex]\( \overrightarrow{AB} \)[/tex]:
[tex]\[ D = C + \overrightarrow{AB} = (4, -1) + (3, -1) = (4 + 3, -1 - 1) = (7, -2) \][/tex]
Alternatively, you could use:
[tex]\[ D = B + \overrightarrow{AC} = (1, 3) + (6, -5) = (1 + 6, 3 - 5) = (7, -2) \][/tex]
However, this result (7, -2) does not match any of the given options in the question. Let’s recheck another method by midpoint property:
### Step 4: Alternative Method Using Midpoints
In a parallelogram, the diagonals bisect each other. So if M is the midpoint of both diagonals AC and BD:
[tex]\[ M = \frac{A + C}{2} = \frac{(-2, 4) + (4, -1)}{2} = \frac{(2, 3)}{2} = (1, 1.5) \][/tex]
Given M should also be midpoint of BD:
[tex]\[ M = \frac{B + D}{2} \implies (1, 1.5) = \frac{(1, 3) + D}{2} \][/tex]
Solving for D:
[tex]\[ 2 \times (1, 1.5) = (1, 3) + D \][/tex]
[tex]\[ (2, 3) = (1, 3) + D \][/tex]
So,
[tex]\[ D = (2, 3) - (1, 3) = (1, 0) \][/tex]
The correct coordinates of point D are:
[tex]\[ \boxed{(1, 0)} \][/tex]
Thus, the correct option is:
D. (1,0)
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