Question

Диагонали равнобедренной трапеции взаимно перпендикулярны. Найти площадь, если основания равны 12 и 16 см.

Answer 1

Нижнее основание АД=16, верхнее БС=12. Точка О-пересечение диагоналей. БО=СО, АО=ДО. 

 АД^2=АО^2+ДО^2. АО=ДО=8*$\sqrt{2}$

 БС^2=БО^2+СО^2.  БО=СО=6*$\sqrt{2}$

 

Sабо=$\frac{1}{2}*8\sqrt{2}*6\sqrt{2}*$=48. Т.к. таких треугольника 2, то их S=96.

Sбос= $\frac{1}{2}*6\sqrt{2}*6\sqrt{2}*$=36

Sаод= $\frac{1}{2}*8\sqrt{2}*8\sqrt{2}*$=64

 

S=96+36+64=196.

Ответ:  196 кв.см. 

Answer 2

S=1/2 a(основание)*H(высоту)...проведи сначало высоту,потом расмотри треугольник который получиться он будет прямоугольный...

Answer 3

Diagonals of an Isosceles Trapezoid

An isosceles trapezoid is a trapezoid with two parallel sides of equal length. In this case, we are given that the diagonals of the trapezoid are mutually perpendicular. We need to find the area of the trapezoid, given that the lengths of the bases are 12 cm and 16 cm.

To find the area of the trapezoid, we can use the formula:

Area = (1/2) * (sum of the lengths of the bases) * (height)

In this case, the sum of the lengths of the bases is 12 cm + 16 cm = 28 cm. We need to find the height of the trapezoid.

To find the height, we can use the fact that the diagonals of the trapezoid are mutually perpendicular. Let's denote the diagonals as AC and BD, where AC is perpendicular to BD. The height of the trapezoid is the distance between the bases, which is equal to the length of the perpendicular from the intersection point of the diagonals to the bases.

Let's denote the intersection point of the diagonals as O. Since the diagonals are mutually perpendicular, we can form right triangles AOC and BOD. The length of AC is equal to the length of BD, as they are diagonals of an isosceles trapezoid. Let's denote the length of AC (or BD) as x.

Using the Pythagorean theorem, we can find the height of the trapezoid:

height = sqrt(x^2 - (12/2)^2)

Since the diagonals are mutually perpendicular, we can use the fact that the product of the lengths of the diagonals is equal to the sum of the squares of the lengths of the bases:

AC * BD = AB^2 + CD^2

Substituting the values, we have:

x * x = 12^2 + 16^2

Simplifying, we get:

x^2 = 144 + 256

x^2 = 400

Taking the square root of both sides, we get:

x = 20

Now we can substitute the value of x into the formula for the height:

height = sqrt(20^2 - (12/2)^2)

height = sqrt(400 - 36)

height = sqrt(364)

height ≈ 19.1 cm

Now we can substitute the values of the bases and the height into the formula for the area of the trapezoid:

Area = (1/2) * (12 cm + 16 cm) * (19.1 cm)

Area ≈ 14.4 cm^2

Therefore, the area of the trapezoid is approximately 14.4 square centimeters.