Question

Меньшая боковая сторона прямоугольной трапеции равна 3 см,а большая образует угол равный 30градусов с одним из оснований.Найти : это основание ,если на нем лежит точка пересечения биссектрис углов при другом основании

Answer 1

Дана прямоугольная трапеция пусть нижнее основание АД верхнее ВС  меньшая боковая сторона АВ  точка пересечения биссектрис на АД- точка М
1) треугольник АВМ.   биссектриса ВМ делит <B пополам  т е по 45* тогда АВ=АМ=3
биссектриса СМ делит  угол С=150* по 75*       <C=180-<D=180-30*=150* тогда трСМД также равнобедренный СД=МД=6 т кСД гипотенуза а катет лежащий против угла 30*  АВ=СН=3 Значит основание АД=3+6=9

Answer 2

Problem Analysis

We are given a trapezoid with a smaller side measuring 3 cm and a larger side forming a 30-degree angle with one of the bases. We need to find the length of that base if the point of intersection of the angle bisectors lies on the other base.

Solution

Let's denote the smaller side of the trapezoid as a and the larger side as b. We need to find the length of the base on which the point of intersection of the angle bisectors lies.

To solve this problem, we can use the properties of a trapezoid and the angle bisectors.

1. The angle bisectors of a trapezoid divide the opposite sides proportionally. This means that the ratio of the lengths of the segments on one base is equal to the ratio of the lengths of the segments on the other base.

2. The angle bisectors of a trapezoid are concurrent, meaning they intersect at a single point.

Let's denote the length of the base on which the point of intersection of the angle bisectors lies as x.

Using the properties mentioned above, we can set up the following equation:

(a + x) / (b - x) = a / b

Simplifying this equation, we get:

a(b - x) = (a + x)b

Expanding and rearranging the equation, we get:

ab - ax = ab + xb

Simplifying further, we get:

ax + xb = ab

Factoring out x, we get:

x(a + b) = ab

Finally, solving for x, we get:

x = (ab) / (a + b)

Now, let's substitute the given values into the equation to find the length of the base:

x = (3 * b) / (3 + b)

Since we know that the larger side forms a 30-degree angle with one of the bases, we can use trigonometry to find the value of b.

In a right triangle, the tangent of an angle is equal to the ratio of the length of the opposite side to the length of the adjacent side. In this case, the opposite side is a and the adjacent side is b.

Using the tangent function, we can write:

tan(30 degrees) = a / b

Substituting the given value of a and solving for b, we get:

b = a / tan(30 degrees)

Now, let's substitute this value of b into the equation for x to find the length of the base:

x = (3 * (a / tan(30 degrees))) / (3 + (a / tan(30 degrees)))

Simplifying this equation will give us the final answer.

Calculation

Let's calculate the length of the base using the given values.

Given: - Length of the smaller side (a) = 3 cm - Angle formed by the larger side with one of the bases = 30 degrees

Using the equation for b:

b = a / tan(30 degrees) = 3 / tan(30 degrees)

Using the equation for x:

x = (3 * (a / tan(30 degrees))) / (3 + (a / tan(30 degrees)))

Substituting the values:

b = 3 / tan(30 degrees) ≈ 5.196 cm

x = (3 * (3 / tan(30 degrees))) / (3 + (3 / tan(30 degrees))) ≈ 2.598 cm

Therefore, the length of the base on which the point of intersection of the angle bisectors lies is approximately 2.598 cm.

Answer

The length of the base on which the point of intersection of the angle bisectors lies is approximately 2.598 cm.