2
$\begingroup$

What is the area of the shaded region of the given $8 \times 5$ rectangle?

enter image description here

Attempt: We know that the area of a kite is $\dfrac{pq}{2}$ where $p$ and $q$ are the diagonals. Thus, since the two lines in the figure intersect at the center of the square, the length of the long diagonal of one of the two shaded kites is half the length of the diagonal of the rectangle, which is $\dfrac{\sqrt{89}}{2}$. Thus the answer should be $\dfrac{\sqrt{89}\sqrt{2}}{2}$ but this is not the right answer (answer is $6.5$). What did I do wrong?

$\endgroup$
  • 1
    $\begingroup$ HINT: They are not kites. $\endgroup$ – K. Jiang Feb 3 '16 at 17:42
  • $\begingroup$ @K.Jiang Why not? $\endgroup$ – user19405892 Feb 3 '16 at 17:44
  • $\begingroup$ The diagonals aren't orthogonal, so it isn't a kite. $\endgroup$ – user302982 Feb 3 '16 at 17:54
2
$\begingroup$

This drawing is misleading, these are not kites indeed. That's the problem.
Make a nice, proportional drawing and you will see these sides there are not equal.

$\endgroup$
  • $\begingroup$ Yes, the sides from the center to the 4 sides of the rectangle. This is not a square. It's a 5x8 rectangle, as you wrote. $\endgroup$ – peter.petrov Feb 3 '16 at 17:58
1
$\begingroup$

Hint:

Each "half" of each shaded quadrilateral is a triangle of base $1$ and known altitude. Note that the altitude is not the same for each "half", so, indeed, they are not kites. (And what I called "half" is not really a half).

$\endgroup$
1
$\begingroup$

We draw a diagonal from the upper-left to the bottom-right corner. This divides the two quadrilaterals into four triangles, two with base $1$ and height $4$ and two with base $1$ and height $\frac{5}{2}.$ Using basic area calculations, the total area should be $\boxed{6.5}.$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.