Question

Сторона прямокутника 4 см і утворює з діагоналлю кут 60°. Знайти цю діагональ.​

Answer 1

Объяснение:

а=4 см

Альфа=60

d=?

cos(aльфа)=а/d

d=a/cos60=4:1/2=4×2=8 cм

Answer 2

Finding the Diagonal of a Rectangle

To find the diagonal of a rectangle, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

In this case, we have a rectangle with one side measuring 4 cm and an angle of 60° formed with the diagonal. We need to find the length of the diagonal.

Let's denote the length of the diagonal as 'd'. According to the Pythagorean theorem, we can set up the following equation:

d^2 = 4^2 + x^2

Where 'x' represents the length of the other side of the rectangle.

To solve for 'x', we need to find the value of 'x' that satisfies the equation. Let's solve it step by step:

1. Square the length of the side of the rectangle: - 4^2 = 16

2. Substitute the squared length of the side and the equation becomes: - d^2 = 16 + x^2

3. We know that the angle formed between the side and the diagonal is 60°. Using trigonometry, we can find the value of 'x' in terms of 'd': - x = d * sin(60°)

4. Substitute the value of 'x' in the equation: - d^2 = 16 + (d * sin(60°))^2

5. Simplify the equation: - d^2 = 16 + d^2 * sin^2(60°)

6. Rearrange the equation to isolate 'd': - d^2 - d^2 * sin^2(60°) = 16

7. Factor out 'd^2': - d^2 * (1 - sin^2(60°)) = 16

8. Simplify the equation using the trigonometric identity sin^2(60°) = 3/4: - d^2 * (1 - 3/4) = 16

9. Simplify further: - d^2 * (1/4) = 16

10. Divide both sides of the equation by 1/4: - d^2 = 16 / (1/4)

11. Simplify the right side of the equation: - d^2 = 16 * 4

12. Calculate the value of 'd': - d^2 = 64 - d = √64 - d = 8 cm

Therefore, the length of the diagonal of the rectangle is 8 cm.