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 use Response 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 ALL 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'll use the Pythagorean theorem.

1. Find the length of diagonal \( CB \): Since \( CB \) is the diagonal across the rectangle with dimensions \( 12 \) (E to C) and \( 4 \) (B to D), we can apply the Pythagorean theorem: \[ CB = \sqrt{12^2 + 4^2} \] \[ CB = \sqrt{144 + 16} = \sqrt{160} = 4\sqrt{10} \approx 12.65 \text{ cm} \] For the response area, you can write:

   • Response area: \( CB = \sqrt{160} \)

2. Find diagonal \( AB \): Now, we will use \( CB \) as one leg of triangle \( ACB \), with \( AC \) being another leg. Since \( AC \) (3) is the height of the prism: \[ AB = \sqrt{CB^2 + AC^2} \] We already calculated \( CB = \sqrt{160} \) and \( AC = 3 \): \[ AB = \sqrt{(\sqrt{160})^2 + 3^2} \] \[ AB = \sqrt{160 + 9} = \sqrt{169} = 13 \text{ cm} \] For the final response area, you can write:

   • Response area: \( AB \approx 13 \text{ cm} \)

So, summarizing the final response areas:

• First Response area (diagonal \( CB \)) = \( \sqrt{160} \)
• Second Response (hypotenuse) for \( AB \) = \( 13 \text{ cm} \)