Question 10 of 13intermediate🔧 ApplyLong Answer4 marks

Three vertices of a parallelogram ABCDABCD are A(3, −1, 2)A(3,\, -1,\, 2), B(1, 2, −4)B(1,\, 2,\, -4) and C(−1, 1, 2)C(-1,\, 1,\, 2). Find the coordinates of the fourth vertex.

Correct Answer

The coordinates of the fourth vertex DD are (1, −2, 8)(1,\, -2,\, 8).

Exercise: Miscellaneous Exercise on Chapter 11 | Q: 1 | (Page Page 8)
For More Understanding

Explanation

This question tests the understanding of properties of a parallelogram in three-dimensional coordinate geometry. The key property used is that the diagonals of a parallelogram bisect each other, which means the midpoint of diagonal ACAC equals the midpoint of diagonal BDBD.

This property is mentioned in the textbook context under Example 7's Note. By equating the coordinates of these midpoints, we can find the unknown coordinates of vertex DD.

Solution Steps

  1. Step 1: Let D(x, y, z) be the fourth vertex of parallelogram ABCD.

  2. Step 2: Apply the property that diagonals of a parallelogram bisect each other. Hence, midpoint of AC = midpoint of BD.

  3. Step 3: Calculate midpoint of AC: ((3 + (-1))/2, (-1 + 1)/2, (2 + 2)/2) = (2/2, 0/2, 4/2) = (1, 0, 2).

  4. Step 4: Calculate midpoint of BD in terms of x, y, z: ((1 + x)/2, (2 + y)/2, (-4 + z)/2).

  5. Step 5: Equate the corresponding coordinates: (1 + x)/2 = 1, (2 + y)/2 = 0, (-4 + z)/2 = 2.

  6. Step 6: Solve for x: 1 + x = 2, so x = 1.

  7. Step 7: Solve for y: 2 + y = 0, so y = -2.

  8. Step 8: Solve for z: -4 + z = 4, so z = 8.

  9. Step 9: Therefore, coordinates of D are (1, -2, 8).

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