Lengths of the sides of quadrilateral

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• anemone
In summary, a quadrilateral is a four-sided polygon with four angles, also known as a quadrangle. It includes shapes such as squares, rectangles, parallelograms, rhombuses, and trapezoids. Its properties include having four sides, four angles, and four vertices, as well as two pairs of parallel sides and opposite angles that add up to 180 degrees. To find the perimeter of a quadrilateral, you add the lengths of all four sides together using the formula P = a + b + c + d. The area of a quadrilateral can be calculated using different formulas depending on the type of quadrilateral, such as multiplying the length and width for a square or rectangle. A quadrilateral is also a paralle
anemone
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The lengths of the sides of a quadrilateral are positive integers. The length of each side divides the sum of the other three lengths. Prove that two of the sides have the same length.

If the quadrilateral has sides $a,b,c,d$ and perimeter $s = a+b+c+d$ then $a < b+c+d$. Also, $a$ divides $b+c+d$, so $b+c+d$ is at least twice $a$. But $b+c+d = s-a$, so $a$ also divides $s$, and $s$ must be at least $3$ times $a$, say $s=ka$ with $k\geqslant3$. Similarly, $s = lb$, $s=mc$ and $s=nd$, with all of $l,m,n$ greater than or equal to $3$.

Then $a = \dfrac1ks$, $b = \dfrac1ls$, $c = \dfrac1ms$, $d = \dfrac1ns$. Therefore $s = a+b+c+d = \dfrac1ks + \dfrac1ls + \dfrac1ms + \dfrac1ns$ and so $$\frac1k + \frac1l + \frac1m + \frac1n = 1.$$ But under the given conditions, if $a,b,c,d$ are all different then so are $k,l,m,n$, and the largest possible value for $\dfrac1k + \dfrac1l + \dfrac1m + \dfrac1n$ is $\dfrac13 +\dfrac14 + \dfrac15 + \dfrac16 = \dfrac{57}{60}$, which is less than $1$. So those conditions cannot be satisfied and therefore at least two of the four sides must have the same length.

1. What is a quadrilateral?

A quadrilateral is a polygon with four sides and four angles.

2. How do you find the perimeter of a quadrilateral?

To find the perimeter of a quadrilateral, you add the lengths of all four sides together.

3. Can a quadrilateral have unequal sides?

Yes, a quadrilateral can have unequal sides. In fact, most quadrilaterals have sides of different lengths.

4. What is the difference between a regular quadrilateral and an irregular quadrilateral?

A regular quadrilateral, also known as a square, has all four sides of equal length and all four angles of equal measure. An irregular quadrilateral, on the other hand, has sides and angles of different lengths and measures.

5. How many different types of quadrilaterals are there?

There are many different types of quadrilaterals, including squares, rectangles, parallelograms, rhombuses, and trapezoids. In total, there are 15 different types of quadrilaterals.

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