Question

В прямоугольном треугольнике биссектриса наименьшего угла пересекает катет под углом 110градусов.Найдите острые углы данного треугольника.Нарисуйте этот треугольник и объяснить.

Answer 1

Угол в = 50; угол = 90 (но он не острый), угол с = 40.. Его делит биссектриса, получается два угла по 20 градусов.. По теореме о том, что сумма углов треугольника = 180.. Можешь найти оставшиеся углы, только внимательно проведи биссектрису под углом 110, смежный с ним угол = 70

Answer 2

Problem Analysis

We are given a right triangle, and we need to find the acute angles of the triangle when the bisector of the smallest angle intersects one of the legs at an angle of 110 degrees.

Solution

Let's denote the acute angles of the triangle as A, B, and C, with C being the right angle. We are given that the bisector of angle A intersects one of the legs at an angle of 110 degrees. Let's call the point where the bisector intersects the leg as D.

To solve this problem, we can use the angle bisector theorem, which states that in a triangle, the ratio of the lengths of the sides that the angle bisector divides is equal to the ratio of the lengths of the other two sides. In other words, if AD is the bisector of angle A, then:

AD/BD = AC/BC

Since angle A is the smallest angle in the triangle, we know that angle B is larger than angle A. Therefore, AD is the shorter segment of the leg that it intersects.

Now, let's draw the triangle and label the points:

      A
     /|
    / |
   /  |
  /   |
 /    |
D-----B
|     |
|     |
C-----+

We are given that angle ADB is 110 degrees. Since angle ADB is the sum of angles A and B, we can write the equation:

A + B = 110

We also know that angle ADB is the sum of angles ABD and ABD. Since AD is the bisector of angle A, angles ABD and ADB are equal. Therefore:

ABD = ADB = 110/2 = 55

Now, we can use the fact that the sum of the angles in a triangle is 180 degrees to find angle C:

A + B + C = 180

Substituting the values we know:

55 + 55 + C = 180

Simplifying:

110 + C = 180

C = 180 - 110

C = 70

Therefore, the acute angles of the triangle are A = 55 degrees, B = 55 degrees, and C = 70 degrees.

Conclusion

In a right triangle where the bisector of the smallest angle intersects one of the legs at an angle of 110 degrees, the acute angles of the triangle are A = 55 degrees, B = 55 degrees, and C = 70 degrees.