Question

В равнобедренную трапецию, периметр которой равен 40, а площадь равна 80, можно вписать окружность. Найдите расстояние от точки пересечения диагоналей трапеции до её меньшего основания

Answer 1

Проведем высоты трапеции ЕР и ВН.
ЕР=ОЕ+ОР=ВН.
Так как в трапецию можно вписать окружность, то выполняется равенство:
АВ+СD=AD+BC
Периметр равен:
P=AB+CD+AD+BC=40, значит
2АВ=20, АВ=10 (трапеция равнобедренная)
AD+BC=20
S=(AD+BC)/2*ЕР, отсюда
ЕР=2S/(AD+BC)= 2*80/20=8  => ВН=8.
Высота ВН делит основание ВD на два отрезка
АН=(AD-BC)/2 и HD=(AD+BC)/2 (свойство равнобедренной трапеции).
2АН=AD-BC.
Из теоремы Пифагора АН=√(АВ²-ВН²)=√(10²-8²)=6.
Итак,
AD+BC=20
AD-BC=12, значит
AD=16, ВС=4.
Треугольики ВОС и АОD подобны по двум углам (даже по трем!),так как <CAD=<ACB и <BDA=<DBC - внутренние накрест лежащие углы
при параллельных ВС и AD и секущих АС и ВD соответственно.
Коэффициент подобия этих треугольников равен k=ВС/AD=1/4.
Тогда ОЕ/ОР=1/4 (высоты подобных треугольников).
ОР=4*ОЕ. ОЕ+ОР=8. 5*ОЕ=8.
ОЕ=8/5=1,6.
Ответ: искомое расстояние равно 1,6.

Answer 2

Problem Analysis

We are given a trapezoid with a perimeter of 40 and an area of 80. We need to find the distance from the point of intersection of the diagonals to the smaller base of the trapezoid.

Solution

To solve this problem, we can use the formulas for the perimeter and area of a trapezoid.

Let's denote the lengths of the two parallel sides of the trapezoid as a and b, and the distance between them as h. The formula for the perimeter of a trapezoid is:

Perimeter = a + b + 2h We are given that the perimeter is 40, so we can write the equation:

40 = a + b + 2h (Equation 1)

The formula for the area of a trapezoid is:

Area = (a + b) * h / 2 We are given that the area is 80, so we can write the equation:

80 = (a + b) * h / 2 (Equation 2)

We have two equations (Equation 1 and Equation 2) with two unknowns (a and b). We can solve these equations simultaneously to find the values of a and b.

Solving the Equations

Let's solve the equations using substitution.

From Equation 1, we can express a in terms of b and h:

a = 40 - b - 2h (Equation 3)

Substituting Equation 3 into Equation 2, we get:

80 = (40 - b - 2h + b) * h / 2

Simplifying the equation:

160 = (40 - 2h) * h

Expanding and rearranging the equation:

2h^2 - 40h + 160 = 0

Now, we have a quadratic equation. We can solve it using the quadratic formula:

h = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = 2, b = -40, and c = 160.

Solving the quadratic equation, we find two possible values for h:

h = 4 or h = 20

Finding the Lengths of the Parallel Sides

Now that we have the value of h, we can substitute it back into Equation 3 to find the lengths of the parallel sides a and b.

For h = 4: a = 40 - b - 2(4) a = 40 - b - 8 a = 32 - b

For h = 20: a = 40 - b - 2(20) a = 40 - b - 40 a = -b

Since the lengths of the sides cannot be negative, we can discard the solution where h = 20.

Therefore, we have h = 4, a = 32 - b, and b = ?.

Finding the Distance from the Intersection of Diagonals to the Smaller Base

To find the distance from the point of intersection of the diagonals to the smaller base, we need to find the length of the smaller base.

Since the trapezoid is isosceles, the lengths of the parallel sides are equal. Therefore, we can set a = b.

Substituting a = b into a = 32 - b, we get:

b = 32 - b

Simplifying the equation:

2b = 32

b = 16

Now we know that the length of the smaller base is 16.

To find the distance from the point of intersection of the diagonals to the smaller base, we can use the formula for the height of a trapezoid:

Height = √(h^2 - ((a - b)^2 / 4)) Substituting the values of h = 4 and a = b = 16 into the formula, we get:

Height = √(4^2 - ((16 - 16)^2 / 4))

Simplifying the equation:

Height = √(16 - 0) = √16 = 4

Therefore, the distance from the point of intersection of the diagonals to the smaller base is 4.

Answer

The distance from the point of intersection of the diagonals of the trapezoid to its smaller base is 4.