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Mathematics

Assertion (A): If the points A(x, 1), B(8, 2), C(9, 4) and D(7, 3) are the vertices of a parallelogram, taken in order, then the value of x is 5.

Reason (R): Diagonals of a parallelogram bisect each other at right angles.

  1. A is true, R is false

  2. A is false, R is true

  3. Both A and R are true

  4. Both A and R are false

Section Formula

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Answer

By mid-point formula,

(x, y) = (x1+x22,y1+y22)\Big(\dfrac{x1 + x2}{2}, \dfrac{y1 + y2}{2}\Big)

If the points A(x, 1), B(8, 2), C(9, 4) and D(7, 3) are the vertices of a parallelogram, taken in order, then the value of x is 5. Reflection, RSA Mathematics Solutions ICSE Class 10.

Midpoint of AC:

MAC=(x+92,1+42)=(x+92,52)M_{AC} = \Big(\dfrac{x + 9}{2}, \dfrac{1 + 4}{2}\Big) = \Big(\dfrac{x + 9}{2}, \dfrac{5}{2}\Big)

Midpoint of BD:

MBD=(8+72,2+32)=(152,52)M_{BD} = \Big(\dfrac{8 + 7}{2}, \dfrac{2 + 3}{2}\Big) = \Big(\dfrac{15}{2}, \dfrac{5}{2}\Big)

Diagonals of //gm bisect each other.

Thus,

Mid-point of AC = Mid-point of BD

⇒(x+92,52)=(152,52)⇒x+92=152⇒x+9=15⇒x=15−9=6.\Rightarrow \Big(\dfrac{x + 9}{2}, \dfrac{5}{2}\Big) = \Big(\dfrac{15}{2}, \dfrac{5}{2}\Big) \\[1em] \Rightarrow \dfrac{x + 9}{2} = \dfrac{15}{2} \\[1em] \Rightarrow x + 9 = 15 \\[1em] \Rightarrow x = 15 - 9 = 6.

So, Assertion (A) is false.

Diagonals of a parallelogram bisect each other, but do NOT necessarily meet at right angles.

They are perpendicular only in special cases like a rhombus or a square.

So, Reason (R) is false.

Both A and R are false.

Hence, Option 4 is the correct option.

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