Дано:

• AB и CD - две отрезка, которые пересекаются в точке O
• AO = OD
• CO = OB

Требуется доказать:

AОS = DОВ

To create a figure that reflects the given triangle ABC through central symmetry with a center point O, follow these steps:

1. Draw a straight line from point A to point O, and then extend it to a point X that is the same distance from point O as A, creating a line segment AX.

2. On the line segment AX, construct a perpendicular line at point O. This line will be the central axis of symmetry.

3. Draw a line from point B to the central axis of symmetry, making sure that the line is perpendicular to the axis.

4. Using a compass, measure the distance from point B to point A, and then draw an arc of the same distance from point X. This arc should intersect the line drawn in the previous step at point Y.

5. The point Y will be the reflected image of point B through central symmetry with center point O.

6. Repeat the previous steps for point C, drawing a line from point C to the central axis of symmetry and then drawing an arc from point X to find the reflected image at point Z.

7. Connect points Y, Z, and O to create a new triangle, which will be the reflected image of triangle ABC through central symmetry with center point O.

So there you have it, a figure that reflects the given triangle through central symmetry with center point O. Now you can impress your classmates with your mathematical skills!

• The triangle CMN is a right triangle, since two of its sides (CM and CN) are perpendicular to each other.
• Using the Pythagorean theorem, we can find the length of the third side, which is CF.
• CF^2 = CM^2 + CN^2 = 4^2 + 3^2 = 16 + 9 = 25
• Therefore, CF = 5 cm.
• Since MF is a side of the right triangle CMF, we can find the length of NF using the Pythagorean theorem again.
• NF^2 = CF^2 + MF^2 = 5^2 + 5^2 = 25 + 25 = 50
• Therefore, NF = √50 ≈ 7.07 cm.

The length of NF is 4.4cm.

Explanation: Using the given information, we can construct a right triangle using CM, MF, and NC as the sides. Since CM and CF are perpendicular, we can use the Pythagorean theorem to find the length of CF. So we have:

CF² = CM² - MF²

CF² = (4cm)² - (5cm)²

CF² = 16cm² - 25cm²

CF² = 9cm²

CF = 3cm

Similarly, we can find the length of CN using the Pythagorean theorem:

CN²= CM² + NC²

CN²= (4cm)² + (3cm)²

CN²= 16cm² + 9cm²

CN²= 25cm²

CN = 5cm

Since we know that MF and NC are perpendicular, we can use the Pythagorean theorem again to find NF:

NF² = MN² + MF²

NF² = (5cm)² + (3cm)²

NF² = 25cm² + 9cm²

NF² = 34cm²

Finally, taking the square root of both sides, we get:

NF = √34cm ≡ 5.83cm ≈ 4.4cm

За теоремой Піфагора в прямокутному трикутнику квадрат довжини гіпотенузи дорівнює сумі квадратів довжин катетів. Тобто, якщо позначити довжину катетів як 'a' та 'b', а гіпотенузи як 'c', то ми отримаємо рівняння c^2 = a^2 + b^2.
У нашому випадку, ми можемо записати таке рівняння: MN^2 = MC^2 + NC^2, де MN - гіпотенуза трикутника MNC, а MC та NC - катети.
Якщо замінити відомі величини, ми отримаємо NF^2 = (4см)^2 + (3см)^2.
Розкривши дужки та скориставшись властивостями степенів, отримаємо NF^2 = 25см^2 + 9см^2 = 34см^2.
Отже, довжина відрізка NF дорівнює квадратному кореню з 34см^2, тобто приблизно 5,83 см.

The perimeter of a rhombus can be calculated by adding all four sides of the shape together. First, we need to find the length of the missing sides using the given diagonal lengths of the rhombus. To do this, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. In this case, the two missing sides of the rhombus form two right triangles with the given diagonal lengths as their hypotenuses.

Let's label the sides of our right triangles as a, b, and c, where a and b are the lengths of the two missing sides and c is the length of the given diagonal. We can set up two equations using the Pythagorean theorem:

a² + b² = 24²
b² + c² = 32²

Solving these equations simultaneously, we get a = 4 and b = 16. Now, we can use the formula for the perimeter of a rhombus, which is P = 4a, where a is the length of one side.

P = 4(4) = 16cm

This means that the perimeter of the rhombus is 16 cm. Don't forget to label your answer with the correct unit of measurement!