Question

Диагонали прямоугольной трапеции взаимно перпендикулярны, и большая из них точкой пересечения делится на отрезки, равные 36 и 34. Определите основание трапеции.

Answer 1

 

извините что то не могу добавить рисунок! треугольники  ВОС и АОД подобны    где точка о пересечения диагоналей трапеций    и кэоффициент  подобия равен   34/36 = 17/18    , так как по условию   трапеция прямоугольная    по тоеоме   пифагора обозначим АО   за    х   тогда  ОС =   17/18 *х            

  как известно      Высота прямоугольного треугольника -среднее геометрическое между проекциями катетов на гипотенузу,

 34^2=x*17/18 *x

  x=6√34

 значит   другая диагональ равна  6√34+6√34*17/18,   теперь      сами основания   

по теореме пифагора  нижнее  равна 

  (6√34)^2 +36^2    =√2520

 верхнее 

 34^2+  (6√34*17/18)^2   ~ 2247 

что то    диагональ какие  то       может неправильно написали!

 

 

Answer 2

Diagonals of a Trapezoid

In a rectangular trapezoid, the diagonals are perpendicular to each other, and the longer diagonal is divided into segments of length 36 and 34. We need to determine the length of the trapezoid's base.

To solve this problem, let's consider the properties of a rectangular trapezoid. A rectangular trapezoid is a trapezoid with two parallel sides and a right angle between them. Let's label the trapezoid as ABCD, with AB and CD being the parallel sides, and AD and BC being the non-parallel sides.

Solution

Let's denote the point of intersection of the diagonals as E. According to the given information, the longer diagonal (AC) is divided into segments of length 36 and 34. Let's label the lengths of these segments as AE = 36 and CE = 34.

To find the length of the trapezoid's base, we need to determine the length of AB or CD.

Using the properties of a rectangular trapezoid, we know that the diagonals are perpendicular, so triangle ADE is a right triangle. We can use the Pythagorean theorem to find the length of AD.

According to the Pythagorean theorem, in a right triangle, the square of the length of the hypotenuse (AC) is equal to the sum of the squares of the lengths of the other two sides (AE and CE).

So, we have:

AD^2 = AE^2 + DE^2

Since DE is equal to CE, we can rewrite the equation as:

AD^2 = AE^2 + CE^2

Substituting the given values, we get:

AD^2 = 36^2 + 34^2

Calculating the right side of the equation:

AD^2 = 1296 + 1156

AD^2 = 2452

Taking the square root of both sides:

AD ≈ 49.52

Now that we have the length of AD, we can find the length of AB or CD by subtracting the lengths of the segments AE and DE from AD.

AB = AD - AE AB ≈ 49.52 - 36 AB ≈ 13.52

Therefore, the length of the trapezoid's base is approximately 13.52.

Conclusion

In a rectangular trapezoid where the diagonals are perpendicular to each other and the longer diagonal is divided into segments of length 36 and 34, the length of the trapezoid's base is approximately 13.52.