The Pythagorean Theorem in Three Dimensions Quick Check

Question

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Question
Use the image to answer the question.

An illustration of a rectangular prism is shown with labeled parts. The left and back base edges are not visible but their positions are indicated by a dashed line. The 6 vertices are labeled A B C D E F and G. The base edge A B is labeled length. Base edge B C is labeled width. The perpendicular C D is labeled height. A dotted line crosses the rectangular base through the center to show diagonal B H. A dashed line extends up from the base indicating the left rear perpendicular E H. A dotted line crosses from the back top left vertex E to the front bottom right vertex B to show diagonal B E. This dotted line is labeled diagonal.

The length of the prism is 32 cm, the width is 24 cm, the height is 44 cm, and the length of the diagonal of the base, segment BH , is 40 cm. Find the length of the diagonal of the rectangular prism, segment BE . Round the answer to the nearest tenth.

(1 point)
Responses

51.2 cm
51.2 cm

46.6 cm
46.6 cm

54.4 cm
54.4 cm

59.5 cm
59.5 cm

Answer 1

To find the length of the diagonal of the rectangular prism (segment BE), we can use the Pythagorean Theorem.

The length of the base (segment AB) is given as 32 cm, the width (segment BC) is given as 24 cm, and the diagonal of the base (segment BH) is given as 40 cm.

We can see that triangle ABH is a right triangle, with segment AB and segment BH as its legs.

Using the Pythagorean Theorem, we have:

AB^2 + BH^2 = BE^2

Substituting the given values:

32^2 + 40^2 = BE^2

1024 + 1600 = BE^2

2624 = BE^2

Taking the square root of both sides to solve for BE:

BE ≈ √2624

BE ≈ 51.2 cm (rounded to the nearest tenth)

Therefore, the correct answer is 51.2 cm.