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3. A square has points [tex]\(A(3, 2)\)[/tex], [tex]\(B(-1, 1)\)[/tex], [tex]\(C(0, -3)\)[/tex], and [tex]\(D(4, -2)\)[/tex]. Identify the coordinates of the point where the diagonals intersect.

Sagot :

GinnyAnswer
To find the intersection point of the diagonals of a square, you can use the midpoint formula. This is because the diagonals of a square bisect each other at their midpoint. Given the points [tex]\( A = (3, 2) \)[/tex], [tex]\( B = (-1, 1) \)[/tex], [tex]\( C = (0, -3) \)[/tex], and [tex]\( D = (4, -2) \)[/tex], we first calculate the midpoints of the diagonals [tex]\( AC \)[/tex] and [tex]\( BD \)[/tex].

### Step 1: Calculate the midpoint of diagonal [tex]\( AC \)[/tex]

The midpoint [tex]\( M_{AC} \)[/tex] of diagonal [tex]\( AC \)[/tex] is given by:
[tex]\[ M_{AC} = \left( \frac{x_1 + x_3}{2}, \frac{y_1 + y_3}{2} \right) \][/tex]
Substitute the coordinates of points [tex]\( A \)[/tex] and [tex]\( C \)[/tex]:

[tex]\[ M_{AC} = \left( \frac{3 + 0}{2}, \frac{2 + (-3)}{2} \right) \][/tex]
[tex]\[ M_{AC} = \left( \frac{3}{2}, \frac{-1}{2} \right) \][/tex]
[tex]\[ M_{AC} = (1.5, -0.5) \][/tex]

### Step 2: Calculate the midpoint of diagonal [tex]\( BD \)[/tex]

The midpoint [tex]\( M_{BD} \)[/tex] of diagonal [tex]\( BD \)[/tex] is given by:
[tex]\[ M_{BD} = \left( \frac{x_2 + x_4}{2}, \frac{y_2 + y_4}{2} \right) \][/tex]
Substitute the coordinates of points [tex]\( B \)[/tex] and [tex]\( D \)[/tex]:

[tex]\[ M_{BD} = \left( \frac{-1 + 4}{2}, \frac{1 + (-2)}{2} \right) \][/tex]
[tex]\[ M_{BD} = \left( \frac{3}{2}, \frac{-1}{2} \right) \][/tex]
[tex]\[ M_{BD} = (1.5, -0.5) \][/tex]

### Step 3: Intersection point of the diagonals

The intersection point of the diagonals of the square is where the two midpoints calculated above coincide. From our calculations, both [tex]\( M_{AC} \)[/tex] and [tex]\( M_{BD} \)[/tex] are the same:
[tex]\[ (1.5, -0.5) \][/tex]

Therefore, the coordinates of the point where the diagonals intersect are:
[tex]\[ \boxed{(1.5, -0.5)} \][/tex]
Hi1315

Answer:

[tex]\left( \frac{3}{2}, \frac{-1}{2} \right)[/tex]

Step-by-step explanation:

  • To find the coordinates of the point where the diagonals of the square intersect, we can use the property that the diagonals of a square bisect each other and meet at the center of the square.
  • The center of the square (the point of intersection of the diagonals) is the midpoint of the diagonals. To find this, we first need to find the midpoints of the diagonals using the given points.

Let's label the points of the square:

-  A(3, 2)

-  B(-1, 1)

-  C(0, -3)

-  D(4, -2)

  • We can pair the opposite points to find the midpoints of the diagonals. We can use the midpoint formula, which states that the midpoint of a line segment connecting points [tex]\text (x_1, y_1) \:and (x_2, y_2) \:is\: given \:by:[/tex]

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

  • First, let's find the midpoint of diagonal AC:

[tex]\text{Midpoint of } AC = \left( \frac{3 + 0}{2}, \frac{2 + (-3)}{2} \right) = \left( \frac{3}{2}, \frac{-1}{2} \right)[/tex]

  • Next, let's find the midpoint of diagonal BD:

[tex]\text{Midpoint of } BD = \left( \frac{-1 + 4}{2}, \frac{1 + (-2)}{2} \right) = \left( \frac{3}{2}, \frac{-1}{2} \right)[/tex]

  • Since both midpoints are the same, we confirm that the diagonals intersect at the point [tex]\left( \frac{3}{2}, \frac{-1}{2} \right).[/tex]

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