Discover How To Find The Area Of A Quadrilateral With Perpendicular Diagonals Using Our Step By Step Formula.

Area of Quadrilateral with Perpendicular Diagonals

A = 1/2(d1)(d2)

To find the area of a quadrilateral with perpendicular diagonals, we can use the formula:

Area = (d1 x d2) / 2

where d1 and d2 are the lengths of the diagonals.

To understand why this formula works, consider the following diagram:

A——B
| |
d1 | | d2
| |
C——D

In the diagram above, imagine that we draw a line segment connecting vertices A and C, and another line segment connecting vertices B and D. These two line segments will intersect at some point, call it E.

Since the diagonals of the quadrilateral are perpendicular, we know that angle AEC and angle BED are both right angles.

Also, since angle AEB is an alternate interior angle with respect to angle AEC, we know that angle AEB is also a right angle.

Therefore, we have a right triangle with sides AE, BE, and d1, and another right triangle with sides DE, BE, and d2.

Using the Pythagorean Theorem, we can write:

AE^2 + BE^2 = d1^2

DE^2 + BE^2 = d2^2

Solving for BE in each equation and setting them equal to each other, we get:

AE^2 + d2^2 – DE^2 = d1^2

Multiplying both sides by AE/2, we get:

(AE/2)(AE^2 + d2^2 – DE^2) = (AE/2)d1^2

The left-hand side of the equation above is just the area of triangle AEB, while the right-hand side is (d1 x d2) / 2.

Similarly, if we use the altitude from vertex B to segment AC, we can show that the area of triangle BEC is also (d1 x d2) / 2.

Therefore, the total area of the quadrilateral is just the sum of the areas of the two triangles, which is:

Area = (d1 x d2) / 2 + (d1 x d2) / 2

Area = (d1 x d2)

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