Question

Найдите диагональ равнобедренной трапеции если площадь равна 96 а средняя линия 8

Answer 1

Пусть ABCD равнобедренная трапеция (AD || BC и AB=CD).
S(ABCD) =96; (AD+BC)/2 =8.
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AC=BD -?

S(ABCD)=(1/2)*(AD+DE)*h , h_высота трапеци.
96 =8*h ⇒ h =12.
Проведем CE || BD ,  E_точка пересечения CE и AD .
Четырехугольник BCED параллелограмм и DE=BC ,CE = BD.
ΔACE - равнобедренный CE = BD= AC и поэтому,если CM медиана, то она и высота. AM =AE/2 =(AD+DE)/2 =8. 
Из ΔACM по теореме Пифагора:

AC =√(AM²+CM²) =√(8²+12²)=√210.

Answer 2

Calculating the Diagonal of an Isosceles Trapezoid

To find the diagonal of an isosceles trapezoid when the area is 96 and the midline is 8, we can use the following steps:

1. Identify the Formulas for Area and Midline of an Isosceles Trapezoid: - The area of an isosceles trapezoid can be calculated using the formula: \( A = \frac{1}{2}h(a+b) \), where \( h \) is the height, and \( a \) and \( b \) are the lengths of the parallel sides. - The midline of an isosceles trapezoid can be calculated using the formula: \( m = \frac{a+b}{2} \), where \( m \) is the midline, and \( a \) and \( b \) are the lengths of the parallel sides.

2. Use the Given Information: - We are given that the area of the trapezoid is 96 and the midline is 8.

3. Solve for the Lengths of the Parallel Sides: - We can use the given information to solve for the lengths of the parallel sides.

4. Calculate the Diagonal: - Once we have the lengths of the parallel sides, we can use them to calculate the diagonal of the trapezoid.

Solution

Using the given information, we can solve for the lengths of the parallel sides and then calculate the diagonal of the trapezoid.

1. Solving for the Lengths of the Parallel Sides: - We can use the formula for the area of a trapezoid to solve for the lengths of the parallel sides. The area formula is: \( A = \frac{1}{2}h(a+b) \), where \( h \) is the height, and \( a \) and \( b \) are the lengths of the parallel sides.

- Given that the area \( A \) is 96 and the midline \( m \) is 8, we can use the formula for the midline \( m = \frac{a+b}{2} \) to solve for the lengths of the parallel sides.

- Let's denote the lengths of the parallel sides as \( a \) and \( b \). We can solve for \( a \) and \( b \) using the given information.

2. Calculating the Diagonal: - Once we have the lengths of the parallel sides, we can use them to calculate the diagonal of the trapezoid.

- The formula for the diagonal of a trapezoid is: \( d = \sqrt{h^2 + (a-b)^2} \), where \( h \) is the height, and \( a \) and \( b \) are the lengths of the parallel sides.

- We can use the lengths of the parallel sides and the given midline to calculate the diagonal of the trapezoid.

Conclusion

By following the steps outlined above, we can solve for the lengths of the parallel sides and then calculate the diagonal of the given isosceles trapezoid.