## Trapezoid

Once I already considered the formula for the diagonals of a trapezoid. When moving from a trapezoid to a rectangle, I ended up with an indefinite expression - zero divided by zero under the square root sign.

Now I propose to consider another formula. In the 7th century, the Indian mathematician Bhaskara I derived a formula for determining

Trapezoid and formula |

## Rectangle

If you apply the conditions describing a rectangle to this formula, you can get a very interesting result of dividing zero by zero:

Formulas for converting a trapezoid into a rectangle |

Zero divided by zero equals zero. I am categorically against this result. Once I used a similar trick in another formula and it turned out that zero divided by zero is equal to one. I agree with this result. But. Never trust anyone, not even yourself, you can make mistakes too. Then I came up with the following idea.

**The result of dividing zero by zero depends on the mathematical operation in which it occurs. When multiplying, the result is equal to one, when adding, the result is zero.**

For the future, you will need to remember this and not be surprised by anything.

## Parallelogram

Let's return to our formula for the area of a trapezoid. I remembered another geometric figure whose parallel sides are equal - this is a

Parallelogram and its area |

This formula contains the sine of the angle between the base of the parallelogram and its side. After transforming the formula for the area of a trapezoid, this factor is completely absent.

The formula for the area of a parallelogram is easily converted to the area of a rectangle. The sine of an angle of 90 degrees is equal to 1. The third factor in the formula for the area of a parallelogram disappears, we get the formula for the area of a rectangle.

## Division by zero

Such an obvious bug in the formula for the area of a trapezoid suggests that it is strictly not recommended to consider the result of dividing zero by zero as correct. The formula for the area of a trapezoid can only be applied to a trapezoid and is not applicable beyond the boundaries of the trapezoid.

After some thought, I found a second option that confirmed my guess about the result of dividing zero by zero. We converted the formula for the area of a trapezoid into the formula for the area of a rectangle. After this, we can easily convert the formula for the area of a rectangle into the formula for the area of a parallelogram. Here's how it's done.

Converting the area of a rectangle to the area of a parallelogram |

Now everything has fallen into place. I take my hat off to the Indian mathematician who teaches us how to correctly understand division by zero.

Don't be afraid to test the math. You will find a lot of interesting things for yourself. Mathematics from a complex science will turn into an ordinary tool of knowledge for you.

In conclusion, I can repeat the conclusion from the previous article: cunning tricks of mathematicians can lead to false results. For an example, see my article “Permuting Addends in Infinite Sums”.

*More interesting math ideas on the page "My Math"*