Question 14 of 18intermediate🔧 ApplyNumerical3 marks

If (1, 2), (4, y), (x, 6) and (3, 5) are the vertices of a parallelogram taken in order, find x and y.

Correct Answer

x = 6 and y = 3

Exercise: EXERCISE 7.2 | Q: 6 | (Page 13)
For More Understanding

Explanation

This question is similar to Example 10 in the textbook. The key property used is that the diagonals of a parallelogram bisect each other. When vertices are given 'in order' as A(1, 2), B(4, y), C(x, 6), D(3, 5), the diagonals are AC and BD. The midpoint of diagonal AC must equal the midpoint of diagonal BD. This gives us two equations which can be solved for x and y.

Solution Steps

  1. Step 1: Let the vertices be A(1, 2), B(4, y), C(x, 6), D(3, 5) taken in order.

  2. Step 2: Since diagonals of a parallelogram bisect each other, the midpoint of AC = midpoint of BD.

  3. Step 3: Midpoint of AC = ((1 + x)/2, (2 + 6)/2) = ((1 + x)/2, 4)

  4. Step 4: Midpoint of BD = ((4 + 3)/2, (y + 5)/2) = (7/2, (y + 5)/2)

  5. Step 5: Equating the coordinates: (1 + x)/2 = 7/2 and 4 = (y + 5)/2

  6. Step 6: From first equation: 1 + x = 7, so x = 6

  7. Step 7: From second equation: 8 = y + 5, so y = 3

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