Question

Биссектриса, проведенная из вершины прямого угла, пересекает гипотенузу под углом 70. Найдите углы, которые образует с катетами высота, проведенная к гипотенузе.

Answer 1

Решение смотри ниже.

Answer 2

Bisector from the Right Angle and its Intersection with the Hypotenuse

When a bisector is drawn from the right angle of a right triangle, it intersects the hypotenuse at a certain angle. In this case, the bisector intersects the hypotenuse at an angle of 70 degrees.

Angles Formed by the Altitude to the Hypotenuse

To find the angles formed by the altitude drawn to the hypotenuse, we need to consider the properties of right triangles.

In a right triangle, the altitude drawn to the hypotenuse creates two smaller triangles that are similar to the original triangle. This is known as the altitude-on-hypotenuse theorem or the geometric mean theorem.

According to the geometric mean theorem, the length of the segment of the hypotenuse that is adjacent to the altitude is the geometric mean of the lengths of the two segments of the hypotenuse created by the altitude.

Let's denote the angles formed by the altitude with α and β. The angles formed by the altitude with the legs of the right triangle will be complementary to α and β.

Using the geometric mean theorem, we can determine the relationship between the lengths of the segments of the hypotenuse and the lengths of the legs of the right triangle.

Let's denote the length of the segment of the hypotenuse adjacent to the altitude as x. Then, the lengths of the two segments of the hypotenuse created by the altitude will be x/α and x/β.

According to the geometric mean theorem, we have the following relationship:

x/α = α/x x/β = β/x

Simplifying these equations, we get:

x^2 = αβ

Taking the square root of both sides, we get:

x = √(αβ)

Therefore, the length of the segment of the hypotenuse adjacent to the altitude is equal to the square root of the product of the angles formed by the altitude.

Now, let's find the angles α and β.

Since the bisector intersects the hypotenuse at an angle of 70 degrees, we can consider the two smaller triangles formed by the altitude and the bisector.

In one of the triangles, the angle formed by the altitude and one of the legs of the right triangle is 70 degrees. Let's denote this angle as θ.

Since the sum of the angles in a triangle is 180 degrees, we have:

θ + α + 90 = 180

Simplifying this equation, we get:

θ + α = 90

Similarly, in the other triangle, the angle formed by the altitude and the other leg of the right triangle is also 70 degrees. Let's denote this angle as φ.

Again, using the fact that the sum of the angles in a triangle is 180 degrees, we have:

φ + β + 90 = 180

Simplifying this equation, we get:

φ + β = 90

Therefore, the angles α and β are complementary to θ and φ, respectively.

To find the values of α and β, we need more information about the triangle, such as the lengths of the legs or the hypotenuse. Without this additional information, we cannot determine the exact values of α and β.

However, we can conclude that α and β are complementary to the angles formed by the altitude and the legs of the right triangle.

Please provide more information about the triangle if you want to determine the exact values of α and β.