Question

СРОЧНО!!! ДАЮ 50 БАЛЛОВ!!! СДЕЛАТЬ ЧЕРТЕЖ И ПРИВЕСТИ Свойства, ТЕОРЕМЫ!!! В прямоугольном треугольнике биссектриса наибольшего угла пересекает гипотенузу под углом 75 градусов. Найдите острые углы данного треугольника.

Answer 1

Ответ:

Острые углы данного треугольника равны 60° и 30°.

Объяснение:

Требуется найти острые углы данного треугольника.

Дано: ΔАВС - прямоугольный;

∠С = 90°;

СМ - биссектриса.

∠АМС = 75°

Найти: ∠А и ∠В.

Решение:

1. Наибольший угол в прямоугольном треугольнике - прямой.

⇒ ∠АСМ = ∠МСВ = 90° : 2 = 45° (СМ - биссектриса)

2. Рассмотрим ΔАМС.

∠АСМ = 45°; ∠АМС = 75°.

• Сумма углов треугольника равна 180°.

⇒ ∠А = 180° - (∠АСМ + ∠АМС) = 180° - (45° + 75°) = 180° - 120° = 60°

3. Рассмотрим ΔАВС - прямоугольный.

• Сумма острых углов прямоугольного треугольника равна 90°.

⇒ ∠В = 90° - 60° = 30°

Острые углы данного треугольника равны 60° и 30°.

Answer 2

Properties of a Right Triangle

A right triangle is a triangle that has one angle measuring 90 degrees. In a right triangle, the side opposite the right angle is called the hypotenuse, and the other two sides are called the legs.

Theorem: Angle Bisector Theorem

The Angle Bisector Theorem states that in a triangle, the angle bisector of an angle divides the opposite side into segments that are proportional to the lengths of the other two sides.

In the given problem, we have a right triangle where the bisector of the largest angle intersects the hypotenuse at an angle of 75 degrees. Let's denote the angles of the triangle as follows:

- The largest angle (opposite the hypotenuse) is denoted as angle A. - The other two angles are denoted as angle B and angle C, with angle B being the angle opposite the leg adjacent to the bisector.

To find the measures of the acute angles B and C, we can use the Angle Bisector Theorem. According to the theorem, the ratio of the lengths of the segments of the hypotenuse formed by the bisector is equal to the ratio of the lengths of the other two sides.

Let's denote the length of the segment adjacent to angle B as x, and the length of the segment adjacent to angle C as y. The length of the hypotenuse is denoted as h.

According to the Angle Bisector Theorem, we have the following proportion:

x/y = h/x

Simplifying the proportion, we get:

x^2 = h * y

Since the bisector intersects the hypotenuse at an angle of 75 degrees, we can use trigonometric ratios to find the values of x and y.

Using the trigonometric ratio for the sine of an angle, we have:

sin(75 degrees) = x/h

Rearranging the equation, we get:

x = h * sin(75 degrees)

Similarly, using the trigonometric ratio for the cosine of an angle, we have:

cos(75 degrees) = y/h

Rearranging the equation, we get:

y = h * cos(75 degrees)

Now we can substitute these values of x and y into the proportion x^2 = h * y to find the relationship between the lengths of the segments.

(h * sin(75 degrees))^2 = h * (h * cos(75 degrees))

Simplifying the equation, we get:

h^2 * sin^2(75 degrees) = h^2 * cos(75 degrees)

Dividing both sides by h^2, we get:

sin^2(75 degrees) = cos(75 degrees)

Taking the square root of both sides, we get:

sin(75 degrees) = cos(75 degrees)

Since sin(75 degrees) = cos(15 degrees), we can conclude that angle B is 15 degrees and angle C is 75 degrees.

Therefore, the acute angles of the given right triangle are 15 degrees and 75 degrees.

Please note that the provided search snippets do not directly address the specific problem of a right triangle with a bisector intersecting the hypotenuse at an angle of 75 degrees. However, the solution provided above is based on the Angle Bisector Theorem and trigonometric ratios, which are fundamental concepts in geometry.