Find the length of the diagonal AB:

Question

The image depicts a rectangular prism labeled with vertices A through H. The dimensions are labeled as follows:
ED (height) = 12
DC (width) = 3
CB (length) = 4
A dashed line represents the space diagonal AB, connecting vertex A at the top to vertex B at the bottom. This prism is likely used to demonstrate the application of the Pythagorean theorem in three dimensions, allowing the calculation of the length of the space diagonal AB. Using the Pythagorean theorem twice, first on the base to find diagonal DB, and then using DB and ED to find AB, the length of AB can be calculated. The search results confirm this as they mention the Pythagorean theorem in 3D.

(5 points)

You would first use Responce Area on the bottom of the box rectangle to find the diagonal CB = Responce Area.

Then us CB as Responce Area of the triangle ACB, with AB being the Responce Area.

AB is approximately = Responce Area.

Responces:
73 cm
3 cm
hypotenuse
Area of a Rectangle
a leg
89 cm
Perimeter of a rectangle
13 cm
Pythagorea Theorem
square root 7
5 cm

Answer 1

To find the length of the diagonal \( AB \) in the rectangular prism, we will apply the Pythagorean theorem.

1. Identify the dimensions:

   • Height \( ED = 12 \) cm
   • Width \( DC = 3 \) cm
   • Length \( CB = 4 \) cm

2. Calculate the diagonal of the base rectangle: The base of the prism is the rectangle formed by sides \( CB \) and \( DC \). Using the Pythagorean theorem, the diagonal \( DB \) can be calculated as follows:

   \[ DB = \sqrt{(CB)^2 + (DC)^2} = \sqrt{(4)^2 + (3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \text{ cm} \]

3. Calculate the space diagonal \( AB \): Now that we have the diagonal \( DB \), we can find the space diagonal \( AB \) using the height \( ED \):

   \[ AB = \sqrt{(DB)^2 + (ED)^2} = \sqrt{(5)^2 + (12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \text{ cm} \]

Thus, the length of the diagonal \( AB \) is 13 cm.