Question

Острый угол равнобокой трапеции равен 45 градусов. Сумма длин ее боковых сторон и меньшего основания равна 18 см.Найти высоту и площадь трапеции если ее диагональ является биссектриссой угла при основании

Answer 1

применены свойства накрест лежащих углов, равнобедренного треугольника, теорема Пифагора

Answer 2

Given Information:

We are given the following information about a trapezoid: - The acute angle of the trapezoid is 45 degrees. - The sum of the lengths of its lateral sides and the smaller base is 18 cm. - The diagonal of the trapezoid is a bisector of the angle at the base.

Solution:

To find the height and area of the trapezoid, we can use the given information and apply some geometric principles.

Let's denote the trapezoid as ABCD, where AB is the larger base, CD is the smaller base, and AD and BC are the lateral sides. Let's also denote the intersection point of the diagonals as O.

Finding the Height:

Since the diagonal of the trapezoid is a bisector of the angle at the base, we can conclude that triangles AOD and BOC are congruent. This is because they share a side (the diagonal) and have two equal angles (45 degrees and 45 degrees).

Therefore, we can say that AO = BO and OD = OC. Let's denote the length of AO (or BO) as x and the length of OD (or OC) as y.

Using the given information that the sum of the lengths of the lateral sides and the smaller base is 18 cm, we can write the equation:

AD + BC + CD = 18

Since AD = AO + OD = x + y and BC = BO + OC = x + y, we can rewrite the equation as:

(x + y) + (x + y) + CD = 18

Simplifying the equation, we get:

2x + 2y + CD = 18

Since we know that the acute angle of the trapezoid is 45 degrees, we can use the properties of a 45-45-90 triangle to find the length of CD.

In a 45-45-90 triangle, the sides are in the ratio 1:1:√2. Since CD is the smaller base, it corresponds to the side of length 1 in the triangle.

Therefore, CD = √2 * y.

Substituting this value into the equation, we get:

2x + 2y + √2 * y = 18

Simplifying further, we have:

2x + (2 + √2) * y = 18

Finding the Area:

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

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

In this case, the sum of the bases is AB + CD.

Using the information that the acute angle of the trapezoid is 45 degrees, we can find the length of AB.

In a 45-45-90 triangle, the sides are in the ratio 1:1:√2. Since AB is the larger base, it corresponds to the side of length √2 in the triangle.

Therefore, AB = √2 * x.

Substituting the values of AB and CD into the formula for the area, we get:

Area = (1/2) * (√2 * x + √2 * y) * y

Simplifying further, we have:

Area = (√2/2) * (x + y) * y

Solving the Equations:

We now have two equations:

1. 2x + (2 + √2) * y = 18 2. Area = (√2/2) * (x + y) * y

To find the values of x, y, height, and area, we need to solve these equations simultaneously.

Unfortunately, the given information does not provide enough information to solve for the values of x, y, height, and area. We would need additional information or equations to find the specific values.

However, we can provide a general solution method and explain the steps involved.

To solve the equations, we can use substitution or elimination methods. We can rearrange equation 1 to solve for x in terms of y:

2x = 18 - (2 + √2) * y x = (18 - (2 + √2) * y) / 2

Substituting this value of x into equation 2, we can express the area in terms of y:

Area = (√2/2) * ((18 - (2 + √2) * y) / 2 + y) * y

Simplifying this equation will give us the area of the trapezoid in terms of y.

To find the height, we can substitute the values of x and y into the equation:

Height = x + y

By substituting the values of x and y into the equation, we can find the height of the trapezoid.

Please note that without specific values for x and y, we cannot provide the exact height and area of the trapezoid.