Question

Диагонали трапеции равны соответственно 9 и 12 см, средняя линия 7.5 см , найти площадь трапеции

Answer 1

Решение на фотографии

Answer 2

Finding the Area of a Trapezoid

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

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

In this case, we are given that the diagonals of the trapezoid are 9 cm and 12 cm, and the length of the median line (which is the average of the lengths of the bases) is 7.5 cm.

Let's calculate the area using the given information.

Calculation

First, let's find the lengths of the bases. Since the median line is the average of the lengths of the bases, we can use the formula:

Median Line = (Length of Base 1 + Length of Base 2) / 2

Given that the median line is 7.5 cm, we can set up the equation:

7.5 = (Length of Base 1 + Length of Base 2) / 2

Multiplying both sides of the equation by 2 gives us:

15 = Length of Base 1 + Length of Base 2

Now, let's solve for one of the bases. We can substitute the value of the other base into the equation:

Length of Base 1 = 15 - Length of Base 2

Next, we can use the formula for the diagonals of a trapezoid:

Diagonal 1 = (Length of Base 1 - Length of Base 2) / 2

Given that Diagonal 1 is 9 cm, we can set up the equation:

9 = (Length of Base 1 - Length of Base 2) / 2

Multiplying both sides of the equation by 2 gives us:

18 = Length of Base 1 - Length of Base 2

Now, let's solve for Length of Base 1:

Length of Base 1 = 18 + Length of Base 2

We can substitute this value back into the equation for the median line:

15 = (18 + Length of Base 2 + Length of Base 2) / 2

Multiplying both sides of the equation by 2 gives us:

30 = 18 + 2 * Length of Base 2

Subtracting 18 from both sides of the equation gives us:

12 = 2 * Length of Base 2

Dividing both sides of the equation by 2 gives us:

Length of Base 2 = 6 cm

Now, we can substitute this value back into the equation for Length of Base 1:

Length of Base 1 = 18 + 6 cm = 24 cm

Now that we have the lengths of the bases, we can calculate the area using the formula:

Area = (Length of Base 1 + Length of Base 2) * (Height) / 2

Substituting the values we found:

Area = (24 cm + 6 cm) * (Height) / 2

Area = 30 cm * (Height) / 2

Given that the diagonals are perpendicular bisectors of each other, we can use the Pythagorean theorem to find the height:

Height^2 = Diagonal 1^2 - (Median Line / 2)^2

Substituting the values we have:

Height^2 = 9 cm^2 - (7.5 cm / 2)^2

Height^2 = 81 cm^2 - 2.25 cm^2

Height^2 = 78.75 cm^2

Taking the square root of both sides of the equation gives us:

Height = √78.75 cm ≈ 8.87 cm

Now, we can substitute the values we found into the formula for the area:

Area = (24 cm + 6 cm) * (8.87 cm) / 2

Area = 30 cm * 8.87 cm / 2

Area ≈ 133.05 cm^2

Therefore, the approximate area of the trapezoid is 133.05 cm^2.

Please note that the calculations provided are approximate due to rounding.