Question

Менша сторона трикутника дорівнює 4 см. У трикутник вписано коло, яке ділиться на дуги, градусні міри яких відносяться як 3:8:9. Знайдіть невідомі сторони трикутника.

Answer 1

Ответ:

ВС≈10,5см

АС=7,6см

Объяснение:

360°- повний кут.

3х+8х+9х=360

20х=360

х=360/20

х=18

∪КМ=3*18=54°

∪KE=8*18=144°

∪МЕ=9*18=162°

∠ВAC=180°-∪KM=180°-54=126°

∠АВС=180°-∪KE=180°-144°=36°

∠ВCA=180°-∪ME=180°-162°=18°

теорема синусів

АВ/sin∠BCA=BC/sin∠BAC=AC/sin∠ABC

4/sin18°=BC/sin126°

BC=4*sin126°/sin18°=4*0,809/0,309≈

≈10,5см

4/sin18°=AC/sin36°

AC=4*sin36°/sin18°=4*0,5878/0,309≈

≈7,6см

Answer 2

#### Problem Analysis We are given a triangle with one side measuring 4 cm. The triangle is inscribed in a circle, and the degrees of the arcs formed by the circle are in the ratio of 3:8:9. We need to find the lengths of the unknown sides of the triangle. #### Solution Let's assume the lengths of the unknown sides of the triangle are a, b, and c. We can use the following steps to find their values: 1. **Find the angles of the triangle:** Since the degrees of the arcs formed by the circle are in the ratio of 3:8:9, we can assign angles to the vertices of the triangle in the same ratio. Let's denote the angles as A, B, and C, with A corresponding to the arc with a degree measure of 3, B corresponding to the arc with a degree measure of 8, and C corresponding to the arc with a degree measure of 9. 2. **Apply the Law of Sines:** The Law of Sines states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. We can use this law to find the lengths of the unknown sides. The Law of Sines can be written as: ``` a/sin(A) = b/sin(B) = c/sin(C) ``` Rearranging the equation, we get: ``` a = sin(A) * (b/sin(B)) = sin(A) * (c/sin(C)) ``` 3. **Substitute the known values:** We know that A = 3 degrees, B = 8 degrees, and C = 9 degrees. We also know that sin(A), sin(B), and sin(C) can be calculated using trigonometric functions. 4. **Calculate the unknown side lengths:** Substitute the known values into the equation and solve for the unknown side lengths. Let's calculate the unknown side lengths using the steps outlined above. #### Calculation 1. **Find the angles of the triangle:** - A = 3 degrees - B = 8 degrees - C = 9 degrees 2. **Apply the Law of Sines:** ``` a = sin(A) * (b/sin(B)) = sin(A) * (c/sin(C)) ``` 3. **Substitute the known values:** - A = 3 degrees - B = 8 degrees - C = 9 degrees 4. **Calculate the unknown side lengths:** - a = sin(3) * (b/sin(8)) = sin(3) * (c/sin(9)) Using trigonometric functions, we can calculate the values of sin(3), sin(8), and sin(9). Let's calculate the unknown side lengths using these values. #### Calculation Continued Using a calculator or a trigonometric table, we find the following values: - sin(3) ≈ 0.0523 - sin(8) ≈ 0.1392 - sin(9) ≈ 0.1564 Substituting these values into the equation, we get: - a ≈ 0.0523 * (b/0.1392) ≈ 0.0523 * (c/0.1564) Since we have two unknowns (b and c) and only one equation, we cannot determine the exact values of b and c. However, we can express them in terms of a. Let's solve for b and c in terms of a. #### Calculation Continued - a ≈ 0.0523 * (b/0.1392) ≈ 0.0523 * (c/0.1564) Simplifying the equation, we get: - b ≈ (a * 0.1392) / 0.0523 - c ≈ (a * 0.1564) / 0.0523 Therefore, the lengths of the unknown sides of the triangle are approximately: - b ≈ (a * 0.1392) / 0.0523 - c ≈ (a * 0.1564) / 0.0523 Please note that these values are approximate and may vary depending on the accuracy of the trigonometric calculations. I hope this helps! Let me know if you have any further questions.