If the diagonal d of a quadrilateral is doubled and the heights h1 and h2 falling on d are halved, then the area of quadrilateral is __________.
Solution:
If d is the diagonal and h1 and h2 are the heights of the triangles formed by the diagonal then,
Area of quadrilateral(A) = sum of the areas A1 and A2 of two triangles of height h1 and h2 respectively
A = 1/2 × d × h1 + 1/2 × d × h2
If diagonal d is doubled and the heights h1 and h2 are halved then the new area will be
A’ = 1/2 × 2d × h1/2 + 1/2 × 2d × h2/2
A’ = 1/2 × d × h1 + 1/2 × d × h2
Therefore we see that after the changes the new area of the quadrilateral is also equal to the original area of the quadrilateral i.e.
A’ = A
✦ Try This: If the diagonal d of a quadrilateral is doubled and the heights h1 and h2 falling on d are also doubled, then the area of quadrilateral is __________.
If d is the diagonal and h1 and h1 are the heights of the triangles formed by the diagonal then,
Area of quadrilateral(A) = sum of the areas A1 and A2 of two triangles of height h1 and h2 respectively
A = 1/2 × d × h1 + 1/2 × d × h2
A = 1/2 × [ d × h1 + d × h2]
2A = d × [ h1 + h2] (1)
If diagonal d is doubled and the heights h1 and h2 are also doubled then the new area will be
A’ = 1/2 × 2d × 2h + 1/2 × 2d × 2h2
A’ = 2 × d × h1 + 2 × d × h2
A’ = 2 × [d × h1 + d × h2]
A’ = 2 × d × [h1 + h2]
From (1) we know,
2A = d × [ h1 + h2]
Therefore
A’ = 2 × 2A
A’ = 4A
Hence the new area is four times the original area of the quadrilateral.
☛ Also Check: NCERT Solutions for Class 8 Maths Chapter 11
NCERT Exemplar Class 8 Maths Chapter 11 Problem 38
If the diagonal d of a quadrilateral is doubled and the heights h1 and h2 falling on d are halved, then the area of quadrilateral is __________.
Summary:
If the diagonal d of a quadrilateral is doubled and the heights h1 and h2 falling on d are halved, then the area of quadrilateral is same as the original area
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