Acilll yapinnnn

YKS TYT
1

1735974920664386714562
1735974920664386714562720×1280 22.4 KB

2

Kimya said “Coordinate System Puzzle Solution with Parallelogram Configuration”

Answer:

To solve the question about finding the distance from point D (Gülçin’s house) to the origin in the coordinate plane where there’s a parallelogram configuration, we need to analyze the given details in the problem.

Step 1: Understanding the Problem

The question provides a coordinate plane showing four neighbors’ houses forming a parallelogram ABCD. Our task is to find the distance from point D (where Gülçin’s house is located) to the origin (0,0).

The given coordinates are:

  • A (1, 4)
  • B (7, 1)
  • C (12, 3)
  • We need to find D(x, y)

Step 2: Find the Coordinates of Point D

Using the properties of a parallelogram and given points, particularly noting that opposite sides of a parallelogram are equal and parallel, we can find D:

  1. The midpoint formula can help find the center of diagonals, which intersect each other in a parallelogram:

    • Midpoint of AC = Midpoint of BD
    • The midpoint of AC is calculated as:
      \text{Midpoint of AC} = \left(\frac{1 + 12}{2}, \frac{4 + 3}{2}\right) = \left(\frac{13}{2}, \frac{7}{2}\right)
    • The midpoint of BD can similarly be found.

  2. Using the condition Midpoint of AC = Midpoint of BD:

    • Let D = (x, y)
    • Midpoint of D and B (7, 1) also occurs at ((\frac{13}{2}, \frac{7}{2})).

  3. Set up equations:

    \frac{x + 7}{2} = \frac{13}{2} \\ \frac{y + 1}{2} = \frac{7}{2}

    Solving for x and y:

    • (x + 7 = 13) → (x = 6)
    • (y + 1 = 7) → (y = 6)

Therefore, D = (6, 6).

Step 3: Calculate the Distance from D to Origin

Using the distance formula, find the distance from D(6, 6) to the Origin(0, 0).

The distance formula is:

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

Substitute the given values where (x1, y1) = (6, 6) and (x2, y2) = (0, 0):

d = \sqrt{(6 - 0)^2 + (6 - 0)^2} \\ d = \sqrt{6^2 + 6^2} \\ d = \sqrt{36 + 36} \\ d = \sqrt{72} \\ d = 6\sqrt{2}

Final Answer:

The distance from D (Gülçin’s house) to the origin is 6\sqrt{2} cm. Thus, the correct option is B.