NCERT Class X Chapter 7: Coordinate Geometry Exercise 7.2 Question 6
NCERT Class X Chapter 7: Coordinate Geometry
Question:
If (1, 2), (4, y), (x, 6) and (3, 5) are the vertices of a parallelogram taken in order, find x and y.
Given:
The vertices of a parallelogram are A(1, 2), B(4, y), C(x, 6), and D(3, 5), taken in order.
To Find:
The values of \( x \) and \( y \).
Formula:
Midpoint formula:
If the endpoints are \( (x_1, y_1) \) and \( (x_2, y_2) \), then the midpoint is
$$
\left( \frac{x_1 + x_2}{2},\ \frac{y_1 + y_2}{2} \right)
$$
In a parallelogram, the diagonals bisect each other, so their midpoints are equal.
Solution:
Step 1: Let the vertices be \( A(1,2) \), \( B(4,y) \), \( C(x,6) \), \( D(3,5) \) in order. In a parallelogram, the diagonals bisect each other. So, the midpoints of \( AC \) and \( BD \) are equal.
Step 2: Find the midpoint of \( AC \):
$$ \text{Midpoint of } AC = \left( \frac{1 + x}{2},\ \frac{2 + 6}{2} \right ) = \left( \frac{1 + x}{2},\ 4 \right ) $$Step 3: Find the midpoint of \( BD \):
$$ \text{Midpoint of } BD = \left( \frac{4 + 3}{2},\ \frac{y + 5}{2} \right ) = \left( \frac{7}{2},\ \frac{y + 5}{2} \right ) $$Step 4: Equate the midpoints of \( AC \) and \( BD \):
$$ \frac{1 + x}{2} = \frac{7}{2} \quad \text{and} \quad 4 = \frac{y + 5}{2} $$Step 5: Solve for \( x \):
$$ \frac{1 + x}{2} = \frac{7}{2} \implies 1 + x = 7 \implies x = 6 $$Step 6: Solve for \( y \):
$$ 4 = \frac{y + 5}{2} \implies 8 = y + 5 \implies y = 3 $$Result:
The values are \( x = 6 \) and \( y = 3 \).
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