In order to fill out the table, we need to apply some basic knowledge about perpendicular and diagonal lines.

First, let's define what a perpendicular line is. A perpendicular line is a line that intersects another line at a right angle. In the case of AB, we know that it is perpendicular, which means that it will form a right angle with another line.

Next, let's look at what a diagonal line is. A diagonal line is a line that connects two opposite corners of a polygon. For AC, we know that it is a diagonal line connecting two opposite corners of the rectangle ABCD.

Since BC is the projection of the diagonal line AC, we can use this information to find its length. The projection of a line is the distance from the endpoint of the line perpendicular to the other line. In this case, the projection of AC onto BC will form a right triangle. Using the Pythagorean theorem, we can find that BC = √(AC² - AB²).

Now, let's apply this knowledge to fill out the table. For the first row, we know that AB is 24 units and BC is 15 units. Using the Pythagorean theorem, we can find that 15 = √(AC² - 24²). Solving for AC, we get an answer of 27 units.

For the second row, we know that AB is still 24 units, but now BC is 12 units. Using the Pythagorean theorem again, we can find that 12 = √(AC² - 24²). Solving for AC, we get an answer of 21 units. And for the last row, we know that AB is 4a and BC is 5a, so by substituting those values into the Pythagorean theorem, we can find that 5a = √(AC² - (4a)²). Solving for AC, we get an answer of 3√5a units.

So now we have filled out all the values for AC, BC, and AB in the table.

The height of a right triangular prism that can inscribe a sphere is equal to half the length of the cube's diagonal.

Using the Pythagorean theorem, we can find that the length of the cube's diagonal is equal to 2√3. Therefore, the height of the prism would be 1√3 or approximately 1.73 units.

• 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.