Question

Диагональ равнобокой трапеции является биссектрисой острого угла разность её оснований 4 см а перниметр 40 см.Найдите среднюю линию трапецию

Answer 1

АВСЕ - трапеция
АВ = СЕ
МК - средняя линия
АС - биссектриса ∠ВАЕ
АЕ - ВС = 4 см
Р(АВСЕ) = 40 см
Найти: МК

Пусть ВС = х, АЕ = у

так как АС - биссектриса ∠ВАЕ, то ∠ВАС = ∠ЕАС 
АЕ || ВС (т.к АВСЕ - трапеция), следовательно ∠АСВ = ∠ЕАС (накрест лежащие углы при параллельных прямых и секущей), отсюда:
∠АСВ = ∠ВАС ⇒ ΔАВС равнобедренный ⇒ АВ = ВС = х

Трапеция равнобокая ⇒ СЕ = АВ = х

Составим систему:
$\left \{ {{y-x=4} \atop {y+3x=40}} \right. \ \ \to \ \ \left \{ {{y=x+4} \atop {x+4+3x=40}} \right.\ \ \to \ \ \left \{ {{y=x+4} \atop {4x=36}} \right.\ \ \to \ \ \left \{ {{y=x+4} \atop {x=9}} \right.\ \ \to \ \ \left \{ {{y=13} \atop {x=9}} \right.$

Средняя линия трапеции равна полусумме оснований:
$MK= \frac{x+y}{2}= \frac{9+13}{2}= \frac{22}{2}=11$ cм.

Ответ: 11 см.

Answer 2

Given Information

We are given the following information about a trapezoid: - The diagonal of the trapezoid is the bisector of the acute angle. - The difference between the lengths of the bases is 4 cm. - The perimeter of the trapezoid is 40 cm.

Solution

To find the length of the median of the trapezoid, we need to determine the lengths of the bases first.

Let's assume that the lengths of the bases are a cm and b cm, where a is the longer base and b is the shorter base.

According to the given information, the difference between the lengths of the bases is 4 cm. Therefore, we can write the equation:

a - b = 4 The perimeter of the trapezoid is given as 40 cm. The perimeter of a trapezoid is the sum of all its sides. In this case, the perimeter can be expressed as:

a + b + 2d = 40 where d is the length of the diagonal.

We are also given that the diagonal is the bisector of the acute angle. This means that the trapezoid can be divided into two congruent right triangles. Let's call the length of the median of the trapezoid m cm.

Using the Pythagorean theorem, we can express the length of the diagonal in terms of the lengths of the bases and the median:

d^2 = a^2 + m^2 d^2 = b^2 + m^2 Now, we have a system of equations with three unknowns (a, b, and m) and three equations (equations and or.

To solve this system of equations, we can use substitution or elimination methods. Let's use the substitution method.

From equation we can express a in terms of b:

a = b + 4

Substituting this value of a into equation we get:

(b + 4) + b + 2d = 40

Simplifying the equation, we have:

2b + 2d = 36 Now, we can substitute the value of a in terms of b into equations and:

d^2 = (b + 4)^2 + m^2 d^2 = b^2 + m^2 Since equations and are both equal to d^2, we can equate them:

(b + 4)^2 + m^2 = b^2 + m^2

Expanding and simplifying the equation, we get:

b^2 + 8b + 16 = b^2

Simplifying further, we have:

8b + 16 = 0

Solving for b, we find:

b = -2

Since the length of a side cannot be negative, we discard this solution.

Therefore, there is no valid solution for the lengths of the bases and the median of the trapezoid that satisfy the given conditions.

In conclusion, there is no trapezoid that meets the given criteria of having a diagonal as the bisector of the acute angle, a difference of 4 cm between the bases, and a perimeter of 40 cm.

Please let me know if you need any further assistance.