a,b,x,y are all numbersProve It said:I can tell already this is not possible, as the LHS is a number while the RHS is a vector...
Albert said:a,b,x,y are all numbers
(they are the length of respective segment)
the RHS is also a number
here $"\times"$
consider it as (a+b) multiplied by (x+y)
[sp]Write $\theta,\phi,\alpha,\beta$ for the angles $ABC,CDA,DAB,BCD$ respectively. Then $$S = \text{area of triangle } ABC + \text{area of triangle } ACD = \tfrac12ay\sin\theta + \tfrac12bx\sin\phi,$$ and also $$S = \text{area of triangle } ABD + \text{area of triangle } BCD = \tfrac12ax\sin\alpha + \tfrac12by\sin\beta.$$ Add those equations to get $$4S = ay\sin\theta + bx\sin\phi + ax\sin\alpha + by\sin\beta.$$ But each of those sines lies between 0 and 1, and so $$4s \leqslant ay + bx + ax + by = (a+b)(x+y).$$ Afterthought: Strictly speaking, that proof only works if the quadrilateral is convex. But a re-entrant quadrilateral obviously has smaller area than the quadrilateral obtained by pushing the re-entrant part out so as to form a convex quadrilateral with the same sides.[/sp]Albert said:Quadrilateral ABCD
given :$\overline{AB}=a ,\overline{BC}=y ,\overline{CD}=b
\,\, and \,\, \overline{AD}=x$
prove :
$4S \leq (a+b)(x+y)$
(where S is the area of ABCD)
A quadrilateral is a 2-dimensional shape with four straight sides and four angles. It is a type of polygon and can have a variety of different shapes and sizes.
The formula for calculating the area of a quadrilateral depends on the type of quadrilateral. For a rectangle or square, you can multiply the length by the width. For a parallelogram, you can multiply the base by the height. For a trapezoid, you can add the lengths of the two parallel sides and multiply by the height, then divide by 2.
No, the formula used to find the area of a quadrilateral depends on the type of quadrilateral. Using the wrong formula will result in an incorrect answer.
A regular quadrilateral has all equal sides and angles, while an irregular quadrilateral has varying sides and angles. Regular quadrilaterals include squares and rectangles, while irregular quadrilaterals can be any other shape.
The area and perimeter of a quadrilateral are related, but not directly. The perimeter is the sum of all the sides of the quadrilateral, while the area is a measure of the space inside the shape. In general, a larger perimeter does not necessarily mean a larger area, as the shape of the quadrilateral can vary greatly.