Question

У прямокутному трикутнику DAB кутA дорівнює 90 градусів, кут D дорівнює 30 градусів, BT - бісектриса трикутника. Знайдіть довжину катета DА, якщо DТ = 20 см.

Answer 1

Ответ:

Объяснение:

Оскільки BT є бісектрисою кута DAB, то кут DBT дорівнює куту ABT, або ж куту BAT (оскільки кут DAB - прямий).

Таким чином, кут DBT = кут ABT = 45 градусів.

Розглянемо правильний трикутник DТВ (де В - середина AB). Оскільки в правильному трикутнику бісектриса розділяє протилежну сторону на дві рівні частини, маємо:

ТВ = ТD = 20/2 = 10 см.

Застосуємо тепер теорему Піфагора до трикутника DAB:

DA^2 = DB^2 + AB^2

З умови задачі відомо, що AB = 2*DB.

Тоді:

DA^2 = DB^2 + (2*DB)^2

DA^2 = 5DB^2

DA = DB\sqrt{5}

Таким чином, довжина катета DA становить:

DA = DB\sqrt{5} = (10\sqrt{3})\sqrt{5} = 10\sqrt{15}  38.7298 см.

Answer 2

Given Information:

- Triangle DAB is a right triangle. - Angle A measures 90 degrees. - Angle D measures 30 degrees. - BT is the bisector of triangle DAB. - DT measures 20 cm.

Solution:

To find the length of side DA, we can use the properties of a right triangle and the angle bisector theorem.

1. Angle Bisector Theorem: - According to the angle bisector theorem, the ratio of the lengths of the two segments created by an angle bisector in a triangle is equal to the ratio of the lengths of the opposite sides. - In triangle DAB, BT is the bisector. Let's denote the length of segment DT as x and the length of segment TA as y. - According to the angle bisector theorem, we have: DT/TA = BD/BA.

2. Right Triangle Properties: - In a right triangle, the sum of the squares of the lengths of the two legs (the sides that form the right angle) is equal to the square of the length of the hypotenuse (the side opposite the right angle). - In triangle DAB, DA is the hypotenuse, and DB and AB are the legs. - We can use the Pythagorean theorem to express the relationship between the lengths of these sides: DB^2 + AB^2 = DA^2.

3. Applying the Angle Bisector Theorem: - Since BT is the bisector, we can use the angle bisector theorem to express the ratio of DT to TA in terms of the lengths of the opposite sides. - In triangle DAB, we have: DT/TA = BD/BA.

4. Applying the Right Triangle Properties: - Using the Pythagorean theorem, we can express the relationship between the lengths of the sides of triangle DAB: DB^2 + AB^2 = DA^2.

5. Solving for the Length of DA: - We can substitute the values we know into the equations and solve for the length of DA. - From the given information, we know that DT = 20 cm and angle D = 30 degrees. - Since angle A is 90 degrees, angle B is 180 - 90 - 30 = 60 degrees. - We can use trigonometric ratios to find the lengths of DB and AB. - In triangle DAB, we have: - DB/DT = sin(angle B) - AB/DT = sin(angle D) - Substituting the known values, we get: - DB/20 = sin(60 degrees) - AB/20 = sin(30 degrees) - Solving these equations, we can find the lengths of DB and AB. - Once we have the lengths of DB and AB, we can substitute them into the equation DB^2 + AB^2 = DA^2 and solve for the length of DA.

Calculation:

Let's calculate the length of DA using the given information and the steps outlined above.

1. Calculating the lengths of DB and AB: - DB/20 = sin(60 degrees) [[1]] - AB/20 = sin(30 degrees) [[2]] - Solving these equations, we find: - DB ≈ 17.32 cm - AB ≈ 10 cm

2. Calculating the length of DA: - DB^2 + AB^2 = DA^2 - (17.32 cm)^2 + (10 cm)^2 = DA^2 - 299.94 cm^2 + 100 cm^2 = DA^2 - 399.94 cm^2 = DA^2 - Taking the square root of both sides, we get: - DA ≈ 19.99 cm

Answer:

The length of the cathetus DA is approximately 19.99 cm.

[[1]] Source: [WolframAlpha](https://www.wolframalpha.com/input/?i=sin%2860+degrees%29) [[2]] Source: [WolframAlpha](https://www.wolframalpha.com/input/?i=sin%2830+degrees%29)