Question

У прямокутному трикутнику катет, протилежний до кута 60º, дорівнює 24 см. Знайти висоту, проведену до гіпотенузи.​

Answer 1

Ответ:

12 см

Объяснение:

дивитись фотографію

Answer 2

Finding the Height of a Right Triangle

To find the height of a right triangle, we can use the trigonometric relationship between the sides of the triangle. In this case, we are given the length of one of the legs (the side opposite a 60º angle) and we need to find the length of the height (the perpendicular drawn to the hypotenuse).

Let's solve the problem step by step:

1. Given information: - Length of the leg opposite the 60º angle (the side adjacent to the height): 24 cm.

2. Trigonometric relationship: - In a right triangle, the sine of an angle is equal to the ratio of the length of the side opposite the angle to the length of the hypotenuse. - In this case, we can use the sine function to relate the length of the height to the length of the leg opposite the 60º angle.

3. Applying the sine function: - Let's denote the length of the height as 'h' and the length of the hypotenuse as 'c'. - The sine of the 60º angle is equal to the ratio of the height to the hypotenuse: sin(60º) = h/c. - Since we know the length of the leg opposite the 60º angle (24 cm), we can substitute it into the equation: sin(60º) = 24/c.

4. Solving for the height: - To find the length of the height, we need to solve the equation sin(60º) = 24/c for 'h'. - Rearranging the equation, we get: h = c * sin(60º). - We can find the value of sin(60º) using a calculator or reference table. The sine of 60º is equal to √3/2. - Substituting the value of sin(60º) into the equation, we have: h = c * (√3/2). - Simplifying further, we get: h = (c * √3) / 2.

5. Finding the length of the hypotenuse: - To find the length of the hypotenuse, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. - In this case, the hypotenuse is the longest side of the triangle, and the other two sides are the leg opposite the 60º angle (24 cm) and the height (h). - Applying the Pythagorean theorem, we have: c^2 = 24^2 + h^2.

6. Solving for the length of the hypotenuse: - We can substitute the value of h from step 4 into the equation c^2 = 24^2 + h^2. - Simplifying, we get: c^2 = 24^2 + ((c * √3) / 2)^2. - Expanding and simplifying further, we have: c^2 = 576 + (3c^2)/4. - Multiplying through by 4 to eliminate the fraction, we get: 4c^2 = 2304 + 3c^2. - Rearranging the equation, we have: c^2 - 3c^2 = 2304. - Simplifying, we get: -2c^2 = 2304. - Dividing through by -2, we have: c^2 = -1152. - Since the length of a side cannot be negative, we can conclude that there is no real solution for the length of the hypotenuse in this case.

7. Conclusion: - Based on the given information, it seems that there is no real solution for the length of the hypotenuse. Therefore, we cannot determine the length of the height of the right triangle.

Please note that the provided solution assumes a standard right triangle with a 90º angle and two acute angles. If the triangle in question deviates from this standard, the solution may not be applicable.