Question

В ровнобедренной трапеции MNPQ основание равно 16 см боковая сторона равна 4корень из 3 и угол M равен 30градусов .Найти площадь трапеции

Answer 1

Построим высоты NH и РН1. Рассмотрим прямоугольный треуг-ик MHN. Зная, что катет прямоугольного треугольника, лежащий против угла в 30 градусов, равен половине гипотенузы, запишем:
NH=1/2MN=2√3 см.
Найдем МН:
cos30=MH:MN, MH=cos30*MN=√3/2 * 4√3=6 см
НН1=NP=MQ-(MH+QH1)=4 см
S MNPQ= 1/2(NP+MQ)*NH=1/2(4+16)*2√3=20√3 см²

Answer 2

Calculation of the Trapezoid's Area

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

Area = (a + b) * h / 2

where: - a and b are the lengths of the parallel bases of the trapezoid, and - h is the height of the trapezoid.

In this case, we are given that the base of the trapezoid MNPQ is 16 cm and the side length PQ is equal to 4√3 cm. We also know that angle M is 30 degrees.

Let's calculate the area step by step.

1. First, we need to find the length of the other base, MN. Since the trapezoid is isosceles, the lengths of the non-parallel sides are equal. Therefore, MN = PQ = 4√3 cm.

2. Next, we need to find the height of the trapezoid. To do this, we can use trigonometry. Since angle M is 30 degrees, we can use the sine function to find the height. The height is the side opposite to angle M, which is PQ. Therefore, the height h = PQ * sin(M) = 4√3 * sin(30°).

3. Now that we have the lengths of the bases and the height, we can substitute these values into the formula to find the area of the trapezoid.

Let's calculate the area using the given values:

MN = PQ = 4√3 cm

h = PQ * sin(M) = 4√3 * sin(30°)

Area = (MN + PQ) * h / 2

Now, let's calculate the area using the given values:

MN = PQ = 4√3 cm

h = 4√3 * sin(30°) = 4√3 * 0.5 = 2√3 cm

Area = (4√3 + 16) * 2√3 / 2

Simplifying the expression:

Area = (20√3) * (√3) / 2

Area = 20 * 3 / 2

Area = 30 cm²

Therefore, the area of the trapezoid MNPQ is 30 cm².