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Find the length of the diagonal AB:

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)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
You would first useResponse area on the bottom of the box rectangle to find the diagonal CB = Response area

Then use CB as Response area of the triangle ACB, with AB being the Response area .

AB is approximately = Response 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

1 answer

To find the length of the diagonal AB of the rectangular prism, we can use the Pythagorean theorem in two steps.

  1. First step: Calculate the diagonal DB on the base of the prism, which is rectangle DBC. Use the dimensions DC and CB:

    • DC (width) = 3 cm
    • CB (length) = 4 cm

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

  2. Second step: Now calculate the space diagonal AB using DB and the height ED:

    • DB (diagonal on the base) = 5 cm
    • ED (height) = 12 cm

    Again use the Pythagorean theorem: \[ AB = \sqrt{(DB)^2 + (ED)^2} = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \text{ cm} \]

Now, we can fill in the responses:

  1. Find the diagonal CB: \( \text{Response area} = 5 cm \)

  2. Use CB as: \( \text{Response area} = \text{hypotenuse} \)

  3. AB is approximately: \( \text{Response area} = 13 cm \)

So, the responses are:

  • Response area for diagonal CB: 5 cm
  • Response area for triangle ACB: hypotenuse
  • Response area for AB: 13 cm