NCERT Class X Chapter 7: Coordinate Geometry Example 10

NCERT Class X Chapter 7: Coordinate Geometry Example 10

Question:

If the points A(6, 1), B(8, 2), C(9, 4) and D(p, 3) are the vertices of a parallelogram, taken in order, find the value of p.

Given:

Points A(6, 1), B(8, 2), C(9, 4), and D(p, 3) are the vertices of a parallelogram ABCD.

To Find:

The value of \( p \).

Formula:

Midpoint formula: The midpoint of a line segment with endpoints \( (x_1, y_1) \) and \( (x_2, y_2) \) is:

$$ \left( \frac{x_1 + x_2}{2},\ \frac{y_1 + y_2}{2} \right) $$

Solution:

Step 1: In a parallelogram, the diagonals bisect each other. Thus, the midpoints of diagonals \( AC \) and \( BD \) must be the same.

Step 2: Find the midpoint of diagonal \( AC \), where \( A(6, 1) \) and \( C(9, 4) \):

$$ \text{Midpoint of } AC = \left( \frac{6 + 9}{2},\ \frac{1 + 4}{2} \right) = \left( \frac{15}{2},\ \frac{5}{2} \right) $$

Step 3: Find the midpoint of diagonal \( BD \), where \( B(8, 2) \) and \( D(p, 3) \):

$$ \text{Midpoint of } BD = \left( \frac{8 + p}{2},\ \frac{2 + 3}{2} \right) = \left( \frac{8 + p}{2},\ \frac{5}{2} \right) $$

Step 4: Since the midpoints are equal, equate their x-coordinates:

$$ \frac{15}{2} = \frac{8 + p}{2} $$

Step 5: Solve for \( p \):

$$ 15 = 8 + p \\ p = 15 - 8 = 7 $$

Result:

The value of \( p \) is 7.

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