Find the length of the diagonal AB:

Question

A three-dimensional rectangular prism (box) is shown. 

The prism's dimensions are labeled: 

* **12** along the edge from E to C.
* **4** along the edge from B to D
* **3** along the edge from C to D

The vertices of the prism are labeled with capital letters (A, B, C, D, E, F, G, H).

A dashed line connects vertex A to the vertex that meets the bottom edge where B, D, and C meet. This dashed line represents a diagonal of the rectangular prism.

(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

Answers for the response areas:
73 cm
3cm
Area of a rectangle
Phythagorean Theorem
5cm
Square root 7
89cm
a leg
13 cm
hypotenuse
perimeter of a rectangle

Answer 1

To find the length of the diagonal \( AB \) in the rectangular prism, we will need to follow the steps outlined in your request.

1. Find the diagonal \( CB \): The points \( C \) and \( B \) are located at coordinates representing their positions in 3D space. Given the dimensions provided, we can find the length of \( CB \) using the Pythagorean Theorem in two dimensions. This uses the edges from \( C \) to \( D \) and \( D \) to \( B \):

   For the rectangle in the plane formed by \( C \) and \( B \):

   • Length from \( C \) to \( D \) (which is 3 cm) is one leg
   • Length from \( D \) to \( B \) (which is 4 cm) is the other leg

   Thus, the diagonal \( CB \) is: \[ CB = \sqrt{(3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \text{ cm} \]

   So, we have:

   • \( CB = \text{Response area} = 5 \text{ cm} \)

2. Find \( AB \) using \( CB \) as one leg of triangle \( ACB \): Now we will treat triangle \( ACB \), where:

   • The length \( AC \) is the height from point \( A \) to point \( C \) and corresponds to the vertical dimension of the prism, which is 12 cm.
   • \( CB \) is the base we just calculated and is 5 cm.

   To find \( AB \), we again use the Pythagorean Theorem: \[ AB = \sqrt{(AC)^2 + (CB)^2} = \sqrt{(12)^2 + (5)^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \text{ cm} \]

Thus, the values we need to fill in according to your instructions are:

• For \( CB \): 5 cm
• For \( AC \): a leg
• For \( AB \): 13 cm

So the responses in their respective areas would be filled as follows:

1. CB: 5 cm
2. \( AC \): a leg
3. AB: 13 cm